Dinah Rosenberg
email: rosenberg@hec.fr

Antoine Salomon
email: salomona@imagine.enpc.fr

Nicolas Vieille
email: vieille@hec.fr.

On Games of Strategic Experimentation

We focus on two-player, two-armed bandit games. We analyze the joint effect on the informational spillovers between the players of the correlation between the risky arms, and the extent to which one's experimentation results are publicly disclosed. Our main results only depend on whethert informational shocks bring good or bad news. In the latter case, there is a sense in which the marginal value of these informational spillovers is zero.

Strategic experimentation issues are prevalent in most situations of social learning. In such setups, an agent may learn useful information by experimenting himself, or possibly, by observing other agents. Typical applications include dynamic R&D (see e.g. Moscarini and Squintani (2010), Malueg and Tsutsui (1997)), competition in an uncertain environment (MacLennan (1984)), financial contracting (Bergemann and Hege (2005) ), etc. We refer to Bergemann and Valimaki (2008) for a recent overview of these applications.

Consider two pharmaceutical firms pursuing research on two related molecules. This research may eventually yield positive results, if the molecule or one of its derivatives turns out to have the desired medical effect, or may never yield such results if this is not the case. In the light of this uncertainty, how long should each firm be willing to wait in the absence of any positive results ? The answer is likely to depend on the informational spillovers between the two firms, on at least two grounds. On the one hand, the correlation between the two research outcomes affects the statistical inferences of the firms. In the above instance, or in, e.g., the case of two firms drilling nearby oil tracts, it is often natural to assume that these outcomes are positively correlated. Klein and Rady (2010) discuss examples from economics of law, in which the relevant outcomes are instead typically negatively correlated. On the other hand, the answer depends on the extent to which research outcomes within one firm are observed by the other firm. There are

cases in which research successes are immediately publicized (say, in the form of specific patents), and other cases in which only research policy decisions are publicly observed.

Our purpose here is to study how these two dimensions -correlation and observabilitycombine and influence the equilibrium outcomes. We use the continuous-time, game-theoretic model of Poisson/exponential bandits, popularized by Keller, Rady and Cripps (2005), and later studied in Keller and Rady (2010), Klein and Rady (2010), Murto and Valimaki (2009) among others. Each player is facing a two-arm bandit problem: at each time instant, each player has to choose whether to stick to a risky arm, or to switch to a safe alternative. The risky arm may be of a bad type, in which case it never delivers any payoff, or of a good type, in which case it starts delivering payoffs at a random time. In contrast with most earlier work, we assume that the decision to switch from the risky arm to the safe one is irreversible. This simplifying assumption allows us to analyze and contrast two scenarios. In one of them, both past decisions and past outcomes are publicly observed; in the second scenario, only past decisions are observed.

The basic strategic insights differ in the two scenarios. Indeed, when both actions and payoffs are observed, there is no asymmetric information between the players, and our game of timing is essentially a game of pure coordination: beyond a certain time, a player is willing to experiment only to the extent that the other player still experiments. Accordingly, the formal analysis is rather straightforward. The probability assigned by a player to his arm being good declines continuously as long as no success occurs. If the two arms are positively correlated, a success hit by the other player brings good news, and the absence of such a success is bad news, so that the beliefs decline faster than in a one-player setup. If the arms are negatively correlated, such an event instead bring bad news, and the belief held by the other player then jumps downwards.

When instead only actions are observed, the game does not exhibit clear strategic complementarities. In any time interval, a player deduces valuable information from the other player's behavior, only inasmuch as that player may choose to drop out in that time interval. There, an event occurs when the other player drops, and this is evidence that he did not get any payoff from the risky arm. The effect of correlation is reversed, when compared to the previous scenario. If the two arms are positively correlated, such an event is bad news. A contrario, such an event is good news if the two arms are negatively correlated.

Our main qualitative finding only hinges on whether events bring good or bad news. When events bring good news, [START_REF] Aghion | Optimal learning by experimentation[END_REF] the belief threshold at which a player chooses to drop out is the same as the belief threshold at which he would optimally choose to drop out, if he were alone. In other words, the (marginal) option value of observing the other player is then equal to zero. In a sense, equilibria thus exhibit no encouragement effect (see Bolton and Harris (1999)). When events instead bring good news, symmetric equilibria always exhibit an encouragement effect, and players keep experimenting with beliefs below the one-player threshold. [START_REF] Bergemann | The financing of innovation: Learning and stopping[END_REF] The proofs for the two observation scenarios are fairly different. Yet, there is a common intuition. If a player is indifferent between dropping out at times t and t + dt, then it must be that the informational benefits derived from experimenting between t and t + dt just offset the opportunity cost of doing so: thus, it might be that an event occurs, that will trigger him to continue beyond t + dt. But if events bring bad news, an event coming from the other player will increase that player's desire to drop out. Thus, the trade-off faced at time t is the same as if one were alone.

When payoffs (and actions) are observed, symmetric equilibria are all pure. If only actions are observed, the unique symmetric equilibrium is mixed, with a continuous distribution, and the cdf of the strategy is obtained as the solution to an linear, second-order differential equation. This difference in equilibrium structure can be understood as follows. At a symmetric equilibrium, there exists no date at which a player expects to 'learn something substantial'. [START_REF] Bergemann | Bandit Problems[END_REF] Otherwise indeed, the marginal value of the information obtained at that date would be positive, and this player would rather wait and get this information, rather than, say, drop out shortly before -this would contradict the equilibrium property. When payoffs are observed, a player 'learns something' from the other when this other player hits a success. This happens according to a continuous distribution. When instead only actions are observed, player i 'learns something' from player j when player j drops out. Any atom in the strategy of player j corresponds to a date at which player i's belief will jump for sure -which cannot happen at a symmetric equilibrium, as we argued. In a sense, the continuity of player j's strategy ensures that the information flow from player j to player i is continuous.

There is no asymmetric equilibrium outcome, unless when arms are negatively correlated and past payoffs are not publicly observed. In that case, there is a flury of such equilibria, and all of them weakly Pareto dominate the symmetric one. All asymmetric equilibria are purely atomic, and the two players never drop out at the same time. Yet, all equilibria share our main qualitative property: when dropping out, the belief held by a player is equal to the belief at which he would drop out in a one-player setup.

Our paper is closely linked to the literature on strategic experimentation, as we indicated above. In the Poisson/exponential setup, Keller, Rady and Cripps (2005), and Klein and Rady (2010) assume that the two risky arms are perfectly correlated (positively in the former case, and negatively in the latter) and analyze the case where payoffs are observed. However, agents may freely switch between the two arms, and these papers are thus not directly comparable to ours. Keller and Rady (2010) study the case where the two risks are perfectly correlated and payoffs observed, and relax the assumption that only a good risky arm may yield payoffs.

To our knowledge, existing work on the case with unobserved payoffs always assumes that the decision to switch to the safe arm is irreversible. Under the assumption that arms are perfectly (positively) correlated Rosenberg et al. (2007) provide qualitative results on the equibrium structure. The present paper is more closely related to Murto and Valimaki (2009). Within a discrete-time framework, they analyze the case in which actions are observed, and assume that risky arms are positively correlated (so that events bring bad news). Their focus is on information aggregation properties at equilibrium, especially in the limit as the number of players goes to +∞. Yet, they argue that, at the symmetric equilibrium, in the continuous-time limit, a property similar to our qualitative observation does hold.

The paper is organized as follows. We lay down the model in Section 1, and state our results in Section 2. Section 3 contains proofs for the observed payoffs case, and an heuristic discussion of the observed actions case. Detailed proofs are in the Appendix.

1 The model 1.1 Setup and comments 1.1.1 The setup Time is continuous. Each of two players is facing a strategic experimentation problem, modeled as a two-arm bandit problem. At each point in time, each player has to choose which of two arms to pull. One of the arms does not involve any uncertainty, and yields a constant payoff flow with present value s. The other arm's type is ex ante unknown, and it may be either Good or Bad. The decision to switch from the risky arm to the safe one is irreversible. That is, each player has to choose when, if ever, to drop out and stop experimenting. We will throughout denote by θ i the time at which player i chooses to drop out.

A risky arm of type Bad (B) never yields any payoff. An arm of type Good (G) yields a constant payoff flow with present value γ, starting from some random time τ , but no payoff prior to τ . The random time τ follows an exponential distribution.

The players have the same prior q over the pair (R 1 , R 2 ) of the types of the risky arms. Conditional on the type profile (R 1 , R 2 ), the two risky arms are independent. That is, in the event where both arms are good, the two success times τ 1 and τ 2 associated with the risky arms of the two players are independent random variables.

We focus on symmetric problems. That is, we assume that (i) the ex ante probability of one's arm being good is the same for both players, (ii) when good, both risky arms have the same characteristics (values of γ and λ) and (iii) players have the same discount rate, r > 0. We make no assumption on the prior q beyond symmetry, and we measure correlation between the two arms by ρ := q(R 1 = R 2 = G)q(R 1 = G)q(R 2 = G). Thus, the two risky arms are positively correlated if ρ > 0, and negatively correlated if ρ < 0.

There is a very simple formal setup fitting this description. Let X 1 and X 2 be two independent random variables with an exponential distribution with parameter λ. Nature chooses the pair (R 1 , R 2 ) of risky types according to the common prior q ∈ ∆({G, B} × {G, B}). The time τ i at which player i's risky arm starts delivering payoffs is

τ i := +∞ if R i = B, and τ i := X i if R i = G.
We will analyze two variants of this model, which differ in the information made available to the players along the play. In both variants, each player i knows at time t (i) his own past payoffs, that is, whether τ i < t or not, and (ii) player j's past actions, that is, whether (and when) player j already dropped out, or not. In one of the two variants, no additional information is provided. In the other variant, player i observes moreover player j's past payoffs -that is, whether τ j < min(θ j , t) or not.

In the latter scenario, actions and payoffs are thus publicly disclosed. Since all information is public, the players always share a common posterior belief over the two types. In this scenario, the game exhibits clear strategic complementarities: the longer player j is willing to experiment, the higher the option value of experimenting for player i and, therefore, the longer player i is willing to experiment. We refer to this scenario, as the (observed ) payoffs scenario.

In the former scenario, only the choices of player j are known to player i, and we refer to it as the (observed ) actions scenario. At time t, the inference made by player i is contingent on player j's strategy: over any time interval [t, t], the 'amount' of information derived by player i from player j depends on how likely it is that player j would drop out between t and t in the absence of a success.

In this scenario, there is no (low-dimensional) state variable. A complete description of a player's beliefs involves the infinite hierarchy of beliefs, and the information interaction is here more complex. As a simple illustration, observe that the optimal payoff of player i is the same, whether player j plans to drop out at time 0, or whether player j plans never to drop out, since in both cases, player i infers no information whatsoever from observing player j. This suggests that, in contrast to the payoffs scenario, player i's optimal payoff is non-monotonic in the amount of experimentation performed by player j.

The one-player benchmark P

One option available to player i is to ignore altogether the information coming from player j. Thus, a natural benchmark is the decision problem P in which player i only observes his own payoffs. The problem P is a standard one-player decision problem . As long as τ i ≥ t, the probability assigned by player i to his risky arm being good, decreases continuously. The unique optimal policy (see Presman (1990) or Presman and Sonin (1990) is to drop out as soon as this belief reaches p * := rs λ(γs) ,

which happens (if no payoff is received) at the time T p defined by

P(R i = G | τ i ≥ T p ) = p * .
As a function of his initial belief p ∈ (0, 1], the optimal payoff of the decision maker in P is given by

W (p) := gp + (s -gp * ) 1 -p 1 -p * (1 -p)p * p(1 -p * ) r/λ if p ≥ p * and W (p) = s if p < p * , where g = +∞ 0 λe -λx e -rx γdx = γ λ r + λ
is the ex ante expected payoff of a good risky arm. To avoid trivialities, we assume g > s > 0. Thus, at equilibrium, no player ever drops out as long as he assigns a probability higher than p * to his arm being good.

For future reference, we also define P as the decision problem in which player i observes at any time t whether τ i < t and/or τ j < t. Equivalently, P is the best-reply problem faced by player i in the payoffs scenario, when player j plans never to drop out.

Good vs. Bad News

The two cases of positive vs. negative correlation are qualitatively different. Consider first the payoffs scenario. As long as players are active, and that no payoff is received, the probability assigned to one's arm being good decreases continuously. It decreases faster than in the benchmark P if ρ > 0, and slowlier if ρ < 0. Indeed, in the former case, (bad) news coming from the other player constitute further statistical evidence that one's arm is bad. When the first success occurs, say at time τ 1 < τ 2 , player 1's arm reveals itself to be good, and player 2 updates his belief. [START_REF] Bolton | Strategic experimentation[END_REF] If ρ > 0, player 1's success is good news, and player 2's belief jumps upwards. If ρ < 0, this is bad news, and player 2's belief jumps downwards.

The statistical inferences in the actions variant are different. As long as both players are active, and as time goes by, the fact that player j does not drop out becomes stronger evidence that player j may have received a payoff from his risky arm. (However, the actual belief held by player i depends upon player j's strategy.) Player i's belief decreases slowlier than in P if ρ > 0, and faster if ρ < 0. If player j drops out, this provides conclusive evidence that τ j ≥ θ j . This is good news for player i if ρ < 0, and bad news if ρ > 0: player i's belief jumps upwards in the former case, and downwards in the latter case.

Both scenarios thus lead to informational shocks. Whether these shocks are interpreted as good or as bad news depends both on the sign of the correlation ρ, and on the informational scenario. As we will see, our main conclusions are mostly driven by shocks bringing good or bad news.

Strategies and outcomes

Since time is continuous, some care is needed when defining strategies. We adopt the viewpoint that a strategy dictates when to stop, as a function of all available information. We argue here that in most cases, the continuation game reduces to a one-player problem. This allows for a straightforward definition of strategies, and of expected payoffs. We treat the two scenarios in turn.

The payoffs scenario

Consider any time instant t, and let us focus on player i, who we assume is active prior to time t. If τ i < t, player i's risky arm is known to be good, and player i should stick to it. On the other hand, if either τ j < t or θ j < t, player j's future behavior does not prove informative to player i. That is, player i should update his belief at time min(τ j , θ j ), and proceed with the optimal policy in the one-player decision problem P. Hence, player i's optimal decision at time t is unambiguous unless if min(τ i , τ j , θ j ) ≥ t. Accordingly, we define a pure strategy of player i to be a time t i ∈ [0, +∞], with the interpretation that player i drops out at time t i if player j has not dropped out, and if no payoff has been received prior to t. A mixed strategy is a probability distribution over [0, +∞].

We now explicit how to compute the expected payoff γ i P (t 1 , t 2 ) induced by an arbitrary pure profile (t 1 , t 2 ).

Assume first that t 1 = t 2 = t. The outcome of the game is determined as follows. In the event where min(τ 1 , τ 2 ) ≥ t, both players drop out at time t. If a success occurs prior to t, say at time τ 1 < min(τ 2 , t), player 2 updates his belief to ψ(τ 1 ), where ψ(x)

:= P(R 2 = G | R 1 = G, τ 2 ≥ x).
Player 2 then remains active until his belief reaches p * , with an expected continuation payoff equal to W (ψ(τ 1 )). Thus, when t 1 = t 2 , the expected payoff of player i is γ i P (t i , t j ) := γE e -rτ i 1 τ i <min(τ j ,t i ) + E e -rτ j W (ψ(τ j ))1 τ j <min(τ i ,t i ) + se -rt i P(min(τ i , τ j ) ≥ t i ) (here and in the sequel, we use without further notice the fact that the probability that τ 1 and τ 2 coincide and be finite is equal to zero:

P(τ 1 = τ 2 < +∞) = 0).
The expected payoff γ i P (t i , t j ) can alternatively be computed as follows. Conditional on R i = B, player i's overall payoff is zero. Next, conditional on R i = G, and R j = B, player i's overall payoff is e -rτ i γ, provided that τ i < t i . The conditional expected payoff (given

R i = G, R j = B) is thus equal to A := t i 0 λe -λx e -rx dx = λ r + λ γ(1 -e -(r+λ)t i ). Finally, conditional
on R i = R j = G, player i's overall payoff is equal to e -rτ i γ, provided that either τ i < t i , or that τ j < t i and τ i ∈ [t i , T ], where T solves ψ(T ) = p * . The conditional expected payoff (given

R i = R j = G) is thus equal to B := λ r + λ γ(1 -e -(r+λ)t i ) + (1 -e -λt i )γ T t i
λe -λx e -rx dx.

Then, γ i P (t i , t j ) is equal to the weighted sum of A and B, using the prior q as weights.

Assume next that t 1 < t 2 . In the event where min(τ 1 , τ 2 ) ≥ t 1 , player 1 drops out at time t 1 , and player 2 next waits until his belief reaches p * . Player 2's continuation payoff at time t 1 is given by W (φ(t 1 )), where φ(x)

:= P(R 2 = G | τ 1 ≥ x, τ 2 ≥ x)
is player 2's belief at time x if no success occurred prior to x. If instead min(τ 1 , τ 2 ) < t 1 , the 'non-successful' player updates his belief at time min(τ 1 , τ 2 ) to ψ(min(τ 1 , τ 2 )), and again, switches to the optimal policy in P, with a continuation payoff equal to W (ψ(min(τ 1 , τ 2 )).

We stress that the outcome of the profile (t 1 , t 2 ) does not depend on t 2 as long as t 2 > t 1 . Observe also that player 1's expected payoff does not depend on t 2 , as long as t 2 ≥ t 1 . Thus, γ 1 P (t 1 , t 2 ) = γ 1 P (t 1 , t 1 ), while

γ 2 P (t 1 , t 2 )
= γE e -rτ 2 1 τ 2 <min(τ 1 ,t 1 ) +E e -rτ 1 W (ψ(τ 1 ))1 τ 1 <min(τ 2 ,t 1 ) +e -rt 1 W (φ(t 1 ))P(min(τ 1 , τ 2 ) ≥ t 1 ).

The actions scenario

We now turn to the definition of strategies and to the computation of payoffs in the actions scenario. Again, consider any time instant t ∈ R + , and let us focus on player i, who we assume remained active prior to t. As above, if τ i < t, player i's arm is known to be good, and player i therefore sticks to it. If now θ j < t, [START_REF] Keller | Strategic experimentation with Poisson bandits[END_REF] this is evidence that τ j ≥ θ j . At time θ j , player i updates his belief to φ(θ j ) (= P(R i = G | min(τ i , τ j ) ≥ θ j )), stays alone and therefore remains active until his belief reaches p * . Hence, player i's optimal decision at t is dictated by P, unless if min(τ i , θ j ) ≥ t. Here again, we thus let a pure strategy of player i be a time t i ∈ [0, +∞], with the interpretation that player i drops out at t i if τ i ≥ t i and θ j ≥ t i . Again, a mixed strategy is a probability distribution over [0, +∞].

Even though a pure strategy is the same mathematical object in the two scenarios, the computation of expected payoffs is of course different. Consider an arbitrary pure profile (t 1 , t 2 ).

If t 1 = t 2 , player i drops out at t i if τ i ≥ t i , and stays forever if τ i < t i . Player i's expected payoff γ i A (t 1 , t 2 ) is thus given by

γ i A (t 1 , t 2 ) := γE e -rτ i 1 τ i <t i .
Assume instead that t 1 < t 2 . Player 1's behavior is as we just described: he drops out at time t 1 if τ 1 ≥ t 1 , and stays forever otherwise. Hence, player 1's payoff is given by

γ 1 A (t 1 , t 2 ) = γE e -rτ 1 1 τ 1 <t 1 .
The payoff of player 2 is to be computed as follows. If player 1 drops out at time t 1 , player 2 updates his belief to φ(t 1 ), and proceeds with the optimal policy in P. If player 1 does not drop out at t 1 , player 2 either drops out at t 2 , or never, according to whether τ 2 ≥ t 2 or τ 2 < t 2 . Hence, player 2's payoff is given by

γ 2 A (t 1 , t 2 ) := e -rt 1 W (φ(t 1 ))P(min(τ 1 , τ 2 ) ≥ t 1 ) + γE e -rτ 2 1 τ 2 <t 1 + 1 τ 1 <t 1 ,τ 2 ∈[t 1 ,t 2 ) .
As in the payoffs scenario, simple integral expressions for these payoffs will be used.

Conceptual issues

The definitions and interpretations of strategies raise a few conceptual issues. For concreteness, let us focus on the actions scenario, and let a pure profile (t 1 , t 2 ) be given, with t 1 < t 2 . According to the previous interpretation, player 2 chooses not to drop out at time t 1 . Assume that the event θ 1 = t 1 < τ 2 materializes. [START_REF] Keller | Strategic experimentation with exponential bandits[END_REF] Again according to our interpretation, player 1 drops out at t 1 , player 2 then updates his belief to φ(t 1 ), and switches to the optimal policy in P. Depending on t 1 and on ρ, it may well be that φ(t 1 ) < p * , in which case player 2 wishes to stop as early as possible following t 1 and even, possibly, to overturn his time t 1 's decision. How then should player 2's decision at time t 1 be defined ? Preventing player 2 to drop out at time t 1 after he has seen that player 1 has exited, would lead to spurious equilibrium non-existence results. As a matter of fact, expected payoffs would not even be defined. [START_REF] Klein | Negatively correlated bandits[END_REF] To avoid this, we implicitly adopted the view that player 2 is allowed to revise his time t 1 decision, should player 1 decide to drop out. This leads to the informal idea that each player has two decision nodes at each time instant t. At the first one, a player conditions his decision on information available prior to t. Should he decide to remain active, and should some informational event occur at time t, he is next allowed to revise his decision -that is, to drop out -, with no delay. [START_REF] Mclennan | Price Diversity and Incomplete Learning in the Long Run[END_REF] We leave aside the technical issues involved in defining a game tree that fits with these interpretations. However, the previous formulas for expected payoffs lead to a well-defined game in strategic form, in which the space of pure strategies is R + ∪ {+∞}.

We emphasize that a strategy only dictates when to stop. That is, the pure strategy t does not specify when to stop, in the counterfactual event where one would not stop at time t when required. This prevents us from discussing equilibrium refinements, and all of our results are stated in terms of Nash equilibrium.

Results

As noted, our qualitative results hinge on the nature of informational shocks -whether they bring good news or bad news. Consequently, we organize the discussion of our results around this criterion.

Three belief functions, φ, ψ and ξ play an important role in the analysis. The first two have already been introduced, and we here recall their definitions. The functions φ and ψ are defined as φ(t)

:= P(R i = G | τ i ≥ t, τ j ≥ t) and ψ(t) := P(R i = G | R j = G, τ i ≥ t).
Thus, φ(t) is the belief held by a player at time t conditional on no success occurring prior to t, while ψ(t) is belief of player i if he knows player j's arm to be good. Note that both φ and ψ are continuous and decreasing. [START_REF] Malueg | Dynamic R&D competition with learning[END_REF] We also set p(t)

:= P(R i = G | τ i ≥ t). Note that ψ(t) > p(t) > φ(t) if ρ > 0, while ψ(t) < p(t) < φ(t) if ρ < 0.
Given a player j's strategy, the function ξ i is defined as

ξ i (t) := P(R i = G | θ j ≥ t, τ i ≥ t).
We stress that ξ i (t) depends on player j's strategy, although this does not appear in the notation. In particular, the continuity and monotonicity properties of ξ i are contingent on player j's strategy. One can check that ξ i is left-continuous, and has a right-limit at each t ∈ R + , which is given by ξ i (t+

) = P(R i = G | θ j > t, τ i > t).
In addition, the map ξ i is decreasing if ρ < 0, irrespective of the strategy of player j.

Negative informational shocks

We start by discussing here the case where events bring bad news. This is the case when either (i) ρ < 0 in the payoffs scenario, or (ii) ρ > 0 in the actions scenario.

Main result

We let dates T p , T φ and T ψ be defined by

p(T p ) = p * , φ(T φ ) = p * and ψ(T ψ ) = p * .
Theorem 1 (payoff scenario, ρ < 0) There is a unique symmetric equilibrium, (T φ , T φ ).

Theorem 2 (actions scenario, ρ > 0) There is a unique symmetric equilibrium, (σ * , σ * ). The distribution σ * has a support10 [T p , T ψ ], and a density, which is positive and continuous on [T p , T ψ ].

One has

ξ i (t) = p * for each t ∈ [T p , T ψ ].
The cdf of σ * is the unique solution of a linear differential equation, see Appendix.

The main conclusion to be drawn is that, at equilibrium, no player is willing to keep experimenting with a belief below p * , the optimal cut-off in P. Put otherwise, while the informational spill-over/externality clearly affects the speed at which beliefs change, the marginal option value of observing the other player is equal to zero, when informational events bring bad news. Indeed, when player i's belief reaches p * , he finds it optimal to exit, whether or not he may benefit from observing the other player's action/payoffs in the future.

While the equilibrium uniqueness claim in Theorems 1 and 2 is partly an artefact of the symmetry requirement, this main conclusion holds at all asymmetric equilibria as well, as we will see below.

There is a simple, yet very partial, intuition behind Theorems 1 and 2. Assume that, at some time t ∈ R + , player i is considering whether to drop out at time t, or to wait an additional, infinitesimal amount of time dt before dropping out. The rationale for waiting an additional dt is that there is a chance that some event will occur that will cause a discontinuity in player i's belief, and trigger him to continue beyond t + dt. Such an event may either be one's own success, or an event coming from the observation of player j. [START_REF] Murto | Learning and information aggregation in an exit game[END_REF] In the latter case however, that event would bring bad news, and would only reinforce player i's desire to drop out. Hence, only the prospect of hitting a success may justify to wait. But then, the condition of being indifferent between exiting at time t or at time t + dt writes the same way, whether in the decision problem P or when player i is observing player j. This sketch abstracts away from a number of complications, and the proofs of Theorems 1 and 2 differ significantly.

Beyond the symmetry requirement

We here provide evidence that our main conclusion does not rely on our restriction to symmetric equilibria. We discuss first the payoffs scenario (with ρ < 0). Note first that φ(t) > p * for t < T φ , hence no player ever stops prior to T φ at equilibrium. We will actually prove that the unique optimal strategy in the decision problem [START_REF] Presman | Poisson version of the two-armed bandit problem with discounting[END_REF] P is to exit at T φ . This will imply that there is a unique equilibrium outcome, in which each player drops out at T φ in the absence of payoffs.

Proposition 1 (payoff scenario, ρ < 0) The equilibria are the profiles (T φ , σ), where σ is any distribution such that σ([T φ , +∞]) = 1, together with the profiles obtained when exchanging the two players.

Since the outcomes induced by (T φ , T φ ) and (T φ , σ) are the same whenever σ([T φ , +∞]) = 1, this multiplicity of equilibria is spurious.

In the actions scenario (with ρ > 0), we do not have a complete characterization of the set of equilibria and, as Proposition 3 below shows, there may be countably many different equilibrium outcomes. However, all the equilibria share the main qualitative feature of the symmetric one.

Proposition 2 (actions scenario, ρ > 0) Let (σ 1 , σ 2 ) be an equilibrium, and let

S i ⊆ R + ∪ {+∞} denote the support of σ i . One has ξ i (t) = ξ i (t+) = p * for each t ∈ S i (i = 1, 2).
Proposition 2 can be interpreted as follows. Consider any date t at which player i may potentially drop out, according to his equilibrium strategy σ i . Then, the belief held by player i, whether computed before observing player j's decision at date t (ξ i (t)) or after observing it (ξ i (t+)) is equal to the one-player optimal threshold, p * .

According to Proposition 3 below, for any k ∈ N, there is an (asymmetric) equilibrium (σ 1 , σ 2 ) in which each of the two strategies assigns positive probability to exactly k dates.

Proposition 3 (actions scenario, ρ > 0) For some specification of the parameters, the following statement is true. For every k ∈ N, there exist two sets

U = {u 1 , . . . , u k } ⊂ R + and V = {v 1 , . . . , v k } ⊂ R + and an equilibrium (σ 1 , σ 2 ) such that • T p = u 1 < v 1 < • • • < u k < v k = T ψ ;
• The supports of σ 1 and σ 2 are respectively equal to U and V.

We believe that the conclusion actually holds true for all parameter values, but leave this outside of the scope of this paper.

Positive informational shocks

We now present our results for the case where events bring good news. This is the case when either (i) ρ > 0 in the payoffs scenario or (ii) ρ < 0 in the actions scenario.

In the payoffs scenario, the game is a game of pure coordination. Player i is willing to experiment beyond T φ only to the extent that player j also experiments beyond T φ . On the other hand, no player will ever be willing to experiment beyond T * * , the (unique) optimal policy in the decision problem P.

As we will show in the Appendix, the optimal strategy in P is

T * * = T φ if ρ < 0.
That is, the optimal belief threshold in the decision problem P is then the same as the optimal belief threshold in P. This explains why there is a unique equilibrium in that case (see Theorem 1 in the previous section).

When ρ > 0, one has T * * > T φ and, not surprisingly, a continuum of equilibrium outcomes.

Theorem 3 (payoff scenario, ρ > 0) One has T * * > T φ . All symmetric equilibria are pure, and all pure equilibria are symmetric. The symmetric equilibria are the pairs

(T, T ), with T ∈ [T φ , T * * ].
The asymmetric equilibria are the profiles (T * * , σ), where σ assigns probability 1 to [T * * , +∞] (together with the profiles obtained when exchanging the two players).

Observe that in all equilibria (with the exception of the equilibrium (T φ , T φ )), players remain active with beliefs below p * .

Conclusions are much sharper in the actions scenario. We introduce the first-order, integrodifferential equation

W (φ(t)) -s λ(γ -s) F ′ (t) = (φ(t) -p * )F (t) + λφ(t)(ψ(t) -p * ) t 0 e λ(t-x) F (x)dx. (1)
Standard results on differential equations imply that there is a unique function, which we denote F * , that solves (1) on the interval [T p , T φ ), and such that F * (t) = 0 for all t ≤ T p .

Theorem 4 (actions scenario, ρ < 0) There is a unique equilibrium, which is a symmetric and non-atomic equilibrium (σ, σ).

The support of the distribution σ is equal to the interval T p , T , where T := min{t : F * (t) = 1} < T φ . The distribution σ has a positive and continuous density on its support, and its cdf is equal to F * on the interval [T p , T φ ].

Since ρ < 0, one has ξ i (t) ≤ p(t) ≤ p * , and the two inequalities are strict for t > T p . That is, in contrast with the case where ρ > 0, players remain active with beliefs below p * .

The proof of Theorem 4 is more delicate. We here discuss some rough intuition behind the result. Beyond the fact that beliefs are strategy-contingent, there is one feature that distinguishes the actions scenario from the payoffs scenario. Assume that player j uses the pure strategy t. When facing this strategy, player i anticipates prior to t, that he will receive a piece of information at t. This obviously creates an incentive for player i to wait until after t, when player j's action choice will become known. This rules out the existence of a pure symmetric equilibrium and, more generally, severely limits the scope for equilibria involving atoms. The reason why equilibria involving atoms fail to exist if ρ < 0, while many such equilibria exist if ρ > 0, is discussed at length in Section 3. As a fact, the most intricate part of the proof of Theorem 4 consists in showing that the cdf's of equilibrium strategies are continuous.

Let us take this conclusion for granted, and let an equilibrium (σ 1 , σ 2 ) be given, with continuous cdf's. For concreteness, let us focus on player 1. Observe first that player 1 keeps being more pessimistic with time, as long as τ 1 ≥ t and θ 2 ≥ t. This is the combined effect of two factors. First, the fact that player 1 does not get any payoff is bad news in itself. Second, the fact that player 2 does not drop out is an increasingly stronger indication that player 2 may have received a payoff with the risky arm, which makes player 2 even more pessimistic since ρ < 0.

On the other hand, the event of player 2 exiting becomes lesser good news with time. Indeed, if player 2 drops out at time t, player 1's continuation payoff jumps to W (φ(t)), and that payoff is decreasing with t.

Since, by the equilibrium condition, player 1 is indifferent between exiting or not at all times in the support of σ, it must be that the chances of observing these good news increase with time. That is, the hazard rate of the exit decision of player 2 should increase with time, in a way that exactly offsets the two effects identified above. This condition writes as Equation (1).

This intuition suggests that the support and the density of σ i are uniquely dictated by the requirement that player j be indifferent between all times in the support of σ j . It also suggests why both strategies σ 1 and σ 2 must have the same, connected, support -hence the uniqueness of the equilibrium.

Proofs

We find it more natural and convenient to present the payoffs scenario first. Most proofs in that case are short and technically elementary, and we thus provide full details. The only one exception is the analysis of the decision problem P, for which technical details are postponed to the Appendix.

By contrast, all proofs in the actions scenario require technical care. Therefore, we will provide detailed sketches here, and will refer the reader to the Appendix for the complete analysis.

The payoffs scenario

The analysis of the payoffs scenario rests upon the analysis of the one-player decision problem P, which we now provide.

The marginal value of observing the other player

We recall that a (pure) policy in P is a time t ∈ R + ∪ {+∞}, with the understanding that the decision maker, say player 1, will choose (i) to drop out at t if min(τ 1 , τ 2 ) ≥ t, (ii) to drop out at t ′ := min{x ≥ t : ψ(x) ≤ p * } if both τ 2 < t ≤ τ 1 and τ 1 ≥ t ′ , and (iii) never to drop out otherwise.

We will show in the Appendix that the decision problem P admits a unique optimal policy, T * * < +∞. We denote by p * * := φ(T * * ) the belief held by the decision maker at time T * * , in the event where min(τ 1 , τ 2 ) ≥ T * * . (Plainly, one has p * * ≤ p * , the optimal threshold in P.)

Proposition 4 One has T * * = T φ if ρ < 0, and T * * > T φ if ρ > 0.
We here limit ourselves with an heuristic proof of Proposition 4. Let us place ourselves at time t := T * * , and assume that min(τ 1 , τ 2 ) ≥ t. By a dynamic programming principle, the decision maker is indifferent between, on the one hand, dropping out at time t and, on the other hand, staying in for an additional, short, amount of time dt, then behaving in an optimal way at time t + dt, in the light of the information acquired between t and t + dt.

There are three events A, B and C to be considered at time t + dt:

A: player 1 hit a success in this additional amount of time (τ 1 ∈ [t, t + dt)),

B: player 2 hit a success but player 1 did not (τ 2 ∈ [t, t + dt), τ 1 ≥ t + dt), C: no one did.
As is usual for continuous-time dynamic programming, we write a first-order expansion of the above indifference condition. The (conditional) probability that A occurs is (approximately) λp * * dt. If event A occurs, the decision maker's overall continuation payoff, (discounted back to time t) is approximately equal to γ. [START_REF] Presman | Sequential control with incomplete information: the Bayesian Approach to multi-armed bandit problems[END_REF] Observe next that the (conditional) probability that both arms yield a payoff between t and t + dt is of the order of (dt) 2 , irrespective of the correlation between the two arms' types. Therefore, the (conditional) probability of B occurring, is approximately equal to the conditional probability that τ 2 ∈ [t, t + dt), which is λp * * dt. If the event B occurs, the flow payoff between t and t + dt is equal to zero, and the continuation payoff of the decision maker is approximately W (ψ(t)).

Finally, the conditional probability that C occurs is the residual probability, 1 -2λp * * dt. On the event C, the flow payoff is exactly 0, while the continuation payoff is exactly s, since the decision maker then drops out at time t + dt. Hence on the event C, the overall payoff is equal to e -rdt s, when discounted back to t.

As a consequence, the first-order expansion of the indifference condition writes

s = λp * * γdt + λp * * W (ψ(T * * ))dt + (1 -rdt) (1 -2λp * * dt) s, which simplifies into λp * * γ + λp * * W (ψ(T * * )) -(r + 2λp * * )s = 0. ( 2
)
If ρ < 0, then ψ(T * * ) < φ(T * * ) ≤ φ(T φ ) = p * , hence W (ψ(T * * )) = s. In that case, equation (2) yields p * * = rs λ(γ -s) = p * ,
and

T * * = T φ as claimed. If ρ > 0, one has ψ(T φ ) > φ(T φ ), hence W (ψ(T φ )) > s,
and T φ is no longer a solution of equation (2). Therefore, T * * > T φ .

Note that, according to equation (2), T * * is a solution to the equation

φ(t) (γ + W (ψ(t)) -2s) = rs λ . (3)
Since the left-hand side of equation ( 3) is the product of the positive and decreasing function φ with a non-increasing function of t, the optimal time T * * is uniquely determined by equation (3).

Proofs of Theorems 1 and 3

We here provide complete proofs of Theorems 1 and 3 for the payoffs scenario. We first describe the best-reply correspondence of either player.

We denote by π the payoff function in the decision problem P. In the light of Section 1.2.1, player i's expected payoff γ i P (t i , t j ) is equal to π(t i ) as long as t i ≤ t j , and does not depend on t i whenever t i > t j .

The continuity properties of γ P (•, •) on the diagonal t 1 = t 2 play a role in the analysis. We fix t 1 ∈ R + , and we compare the outcomes induced by the profiles (t 1 , t 1 ) and (t 1 , t 2 ), where t 2 > t 1 . The two profiles yield the same outcome, except possibly if min(τ 1 , τ 2 ) ≥ t 1 . On the latter event, both players do exit at time t 1 according to the profile (t 1 , t 1 ). Instead, under the profile (t 1 , t 2 ), player 1 drops out at time t 1 , and player 2 remains active until his belief reaches p * , with a continuation payoff equal to W (φ(t 1 )) ≥ s. Formally,

γ 2 P (t 1 , t 2 ) -γ 2 P (t 1 , t 1 ) = P(min(τ 1 , τ 2 ) ≥ t 1 )e -rt 1 (W (φ(t 1 )) -s) .
The continuation payoff W (φ(t

(i) B 2 (t 1 ) = (t 1 , +∞] if t 1 < T φ , (ii) B 2 (t 1 ) = [t 1 , +∞] if t 1 ∈ [T φ , T * * ], and (iii) B 2 (t 1 ) = {T * * } if t 1 > T * * .
In particular, if (t, t) is a pure symmetric equilibrium, then t ≥ T φ by (i) and t ≤ T * * by (ii), hence t ∈ [T φ , T * * ]. Conversely, any profile (t, t) such that t ∈ [T φ , T * * ] is an equilibrium.

Much more precise conclusions are readily available. Observe that the pure strategy T * * is a best reply to any pure strategy of player 1 and, therefore, to any mixed strategy as well. Fix now a mixed strategy σ 1 of player 1, and let t 2 ∈ [0, +∞] be arbitrary. Let us compare the outcomes induced by the two profiles (σ 1 , t 2 ) and (σ 1 , T * * ). On the event min(τ 1 , τ 2 , θ 1 ) < min(t 2 , T * * ), the two outcomes coincide. Assume instead that the event min(τ 1 , τ 2 , θ 2 ) ≥ min(t 2 , T * * ) occurs. If t 2 > T * * , it is strictly suboptimal to continue any longer beyond T * * , hence the expected continuation payoff, conditional on min(τ 1 , τ 2 , θ 1 ) ≥ T * * is strictly higher under (σ 1 , T * * ). Thus, one has γ 2 P (σ 1 , t 2 ) = γ 2 P (σ 1 , T * * ) if and only if σ 1 ((T * * , +∞]) = 0. If now t 2 < T * * , the only optimal policy is to continue until Player 1 drops. Thus, conditional on min(τ 1 , τ 2 , θ 1 ) ≥ t 2 , the expected continuation payoff of player 2 is strictly higher under (σ 1 , T * * ). In that case, γ 2 P (σ 1 , t 2 ) = γ 2 P (σ 1 , T * * ) if and only if σ 1 ((t 2 , +∞]) = 0. This allows us to characterize the set of all equilibria. Since γ i P is not a continuous function, a little care is needed. Let (σ 1 , σ 2 ) be an equilibrium, and denote by S i the support of the distribution σ i . By the equilibrium property, one has

γ i (t i , σ j ) = γ i (σ i , σ j ) (= γ i (T * * , σ j )) for σ i -a.e. t i ∈ S i . Set t i = min S i and ti = max S i . Note that t i ≥ T φ for i = 1, 2.
Assume that, say, t 1 < T * * . [START_REF] Rosenberg | Social learning in one-arm bandit problems[END_REF] Since t 1 ∈ S 1 , there exist strategies t 1 arbitrarily close to t 1 , such that γ 1 (t 1 , σ 2 ) = γ 1 (T * * , σ 2 ). By the previous paragraph, one has σ 2 ((t 1 , +∞]) = 0 for any such t 1 . This implies that t2 ≤ t 1 and, a fortiori, t 2 < T * * . This allows us to exchange the roles of the two players in this argument. One then obtains t1 ≤ t 2 . Hence, t 1 = t1 = t 2 = t2 : (σ 1 , σ 2 ) is a pure symmetric equilibrium.

Assume now that, say, t1 > T * * . The previous argument shows that t 1 , t 2 ≥ T * * . Since t1 ∈ S 1 , there exist strategies t 1 > T * * such that γ 1 P (t 1 , σ 2 ) = γ 1 P (T * * , σ 2 ). Therefore, σ 2 ((T * * , +∞]) = 0.

Hence, σ 2 is the pure strategy T * * , while σ 1 is concentrated on [T * * , +∞]. Conversely, any such pair (σ 1 , σ 2 ) is an equilibrium. This concludes the proof of Theorems 1 and 3, and of Proposition 1 as well.

The actions scenario

We here provide the main insights into the actions scenario, but we postpone all technical details to the Appendix. While we remain at an informal level, we follow rather closely the formal proof, for the reader's convenience. In Section 3.2.1, we gather a number of observations, which hold irrespective of the correlation. We will next discuss the cases of a positive and of a negative correlation in turn.

We recall that a (pure) strategy of player i is a time t ∈ R + ∪ {+∞}, with the interpretation that player i exits at time t if min(τ i , θ j ) ≥ t, and otherwise behaves optimally from time min(θ j , τ i ) on.

For most of the section, we let a tentative equilibrium (σ 1 , σ 2 ) be given, and we proceed using necessary equilibrium conditions. Throughout, we denote by S i the support of the distribution σ i .

Atoms, beliefs and exit

We first discuss the impact on the exit decision of player i, of atoms in the distribution σ j . Recall that, when at time t, player i only knows whether player j dropped prior to t, that is, whether θ j < t. Whether player j drops out at time t will become known immediately after t. As we argue in the Appendix, this implies that when facing σ j , player i's payoff function t → γ i A (t, σ j ) is left-continuous, with a right-limit, γ i A (t+, σ j ) at each t. The belief function t → ξ i (t) enjoys similar continuity properties.

If σ j ({t}) = 0, waiting an infinitesimal amount of time beyond t before possibly dropping out, makes no difference : player i's payoff function γ i A (•, σ j ) is then continuous at t, and the belief ξ i (•) is continuous as well.

If instead there is a positive probability that player j drops out at t, (that is, if σ j has an atom at t), then waiting may make a difference. Indeed, if player j choose to drop out, the continuation payoff of player i is then W (φ(t)) rather than s, had player j dropped out at t. Formally, one has

γ i A (t+, σ j ) -γ i A (t, σ j ) = P(τ i ≥ t, θ j = t) (W (φ(t)) -s) . (4)
Note in particular that, whenever the payoff is discontinuous, it jumps upwards.

The right-continuity of the payoff function readily implies the observation O1 below.

O1 If γ i A (t+, σ j ) > γ i A (t, σ j ), then γ i A (t ′ , σ j ) < γ i A (σ i , σ j ) for every t ′ ≤ t close enough to t.
This observation formalizes the following intuition. Under the assumption in O1, the optimal continuation payoff of player i 15 at time t+ does depend on player 2's decision at time t. And since there is a positive probability that player j stops at t, the information gotten at time t has a positive value. Then, when at time t ′ ≤ t close to t, the continuation payoff of player i exceeds s if he waits until told the decision of player j at time t, and then behaves optimally. Hence it is suboptimal to exit at t ′ .

We next discuss the relation between the belief ξ i (t) held by player i at time t, and his optimal decision at time t.

We first formalize the intuition that, when this belief exceeds p * , it it optimal not to drop out. Assume that ξ i (t+) > p * . Thus, when at time t, and after hearing player j's decision, player i is optimistic -even if he were to be alone, he would find it optimal to experiment. Again, this has a simple consequence. When at time t+, 16 the continuation payoff of player i is higher than s if he chooses to experiment for at least a short additional amount of time, and next to behave optimally. This leads to our second observation.

O2 If ξ i (t+) > p * , then γ i A (t ′ , σ j ) < γ i A (σ i , σ j ) for every t ′ in a neighborhood of t.
We now discuss a bit more formally the extent to which player i should be willing to experiment with beliefs below p * . Assume that ξ i (t) < p * for some t ∈ R + , and assume first that the correlation ρ is positive. We claim that player i would rather drop shortly before t, rather than wait until t. Let us indeed place ourselves at a time t ′ < t very close to t, and assume that player i is considering whether to experiment until t (but not until t+). Since ξ i (t) < p * , if player i were not taking player j's actions into account, waiting until t would yield an expected continuation payoff below s. More precisely, the expected loss would be of the order of tt ′ . 17 Observing player j's decisions in the time interval [t ′ , t] compensates to some extent for these losses. However, the decisions of player j between t ′ and t make a difference to the behavior of player i only if player j happens to drop out in the time interval [t ′ , t). Given that ρ > 0, player i then drops out immediately following player j, rather than waits until t. By dropping out earlier, player i saves the cost of waiting, which is of the order of tt ′ . However, the probability of that event -the probability that player j drops out between t ′ and t -, decreases to zero as t ′ → t. Thus, for t ′ close enough to t, the marginal value of observing player j's actions is arbitrary small compared to tt ′ and therefore, small compared to the expected loss.

15 conditional on τ 1 ≥ t 16 and conditional on player j being active 17 The expected loss also depend on the difference p *ξ i (t). This difference is here kept fixed.

We stress that this conclusion does not carry over to the case where the correlation ρ is negative. Assume indeed now that ρ < 0. If player j drops out at some time x ∈ [t ′ , t), the continuation payoff of player i is then equal to W (φ(x)). If φ(t) > p * , the difference W (φ(x))s is bounded away from zero (for t ′ close to t). Thus, if the probability that player j drops out in the interval [t ′ , t) is itself of the order of tt ′ , the marginal expected gain in observing player j's decisions is of the order of tt ′ , and may well justify experimenting until t.

Nevertheless, there are still two cases in which the above conclusion remains valid. Firstly, if φ(t) ≤ p * , and since W • φ is a C 1 function, the difference W (φ(x))s is at most of the order of tt ′ . Then, exactly as in the case where ρ > 0, the expected marginal gain of observing player j is arbitrarily small compared to tt ′ . Secondly, if the strategy σ j is such that player j cannot possibly stop in [t ′ , t), then the marginal value of observing player j is trivially zero.

This discussion is summarized in the two observations below.

O3 Assume that ξ i (t) < p * and φ(t) ≤ p * . Then γ i A (t, σ j ) < γ i A (σ i , σ j ).

O4 Assume that ξ i (t) < p * and that t / ∈ S j . Then

γ i A (t, σ j ) < γ i A (σ i , σ j ).
The previous observations O1 through O4 have useful immediate consequences, which are valid irrespective of the sign of the correlation.

Claim 5 If t ∈ S i \ S j , then ξ i (t) = p * .
That is, if player i finds it optimal to drop out at t / ∈ S j , the belief of player i at t is equal to p * .

This claim follows immediately from O2 and O4, noting that ξ i (•) and γ i A (•, σ j ) are continuous at t since t / ∈ S j .

Claim 6 Assume that σ j ({t}) > 0. Then σ i ([tε, t]) = 0 for (every) ε > 0 small enough.

That is, if the strategy of player j has an atom at t, player i either stops well before t, or after t. The proof of this claim combines the different observations. If γ i A (t+, σ j ) > γ i A (t, σ j ), the conclusion follows from O1. Assume now that γ i A (t+, σ j ) = γ i A (t, σ j ). In that case, one has φ(t) ≤ p * using (4), and ξ i (t) = ξ i (t+). Note that the belief held by player i at time t+ is either equal to φ(t) if player j drops out, or to ξ i (t+), if player j does not drop out. Since the 'prior' belief ξ i (t) at time t is equal to the expectation of the belief held at time t+ (this is the martingale property of beliefs), it must be that ξ i (t) belongs to the interval [φ(t), ξ i (t+)]. This implies that either ξ i (t) < p * , or ξ i (t+) > p * . The conclusion will follow using O3 in the former case, and from O2 in the latter.

The case of good news

We here assume that the correlation between the two arms is negative. The observations made in the previous section allow us to drastically reduce the range of potential equilibria, as we now show.

Claim 7 Assume ρ < 0. Then S i = S j and the common set is a compact interval of the form [T p , T ], where T is such that φ( T ) > p * . Moreover, none of the two strategies σ i and σ j can possibly have an atom in (T p , T ], and at most one of the two has an atom at T p .

The final assertion on the atoms follows immediately from the other assertions, using Claim 6.

Since ρ < 0, the belief ξ i (t) is decreasing with time. We first argue, by means of contradiction, that S i = S j . Assume instead that t 0 ∈ S i \S j , for some t 0 ∈ R + . By O4, one thus has ξ i (t 0 ) = p * , and by O2, one has t 0 = min S i . That is, player cannot possibly stop prior to t 0 , so that the belief held by player is ξ j (t) = p(t) for every t < t 0 . Since the correlation between the two risky arms is negative, one has p(t) ≤ ξ i (t) for every t < t 0 and therefore, ξ j (t) > p * for every such t. Hence, t 0 < min S j by O2. This implies in turn that p(t 0 ) = ξ i (t 0 )(= p * ).

Next, set t 1 := min{t > t 0 : t ∈ S i ∪ S j }. It is not difficult to combine the previous remarks to show that t 1 > t 0 and t 1 ∈ S i ∩ S j so that t 1 = min S j . Arguing as above, one then has ξ j (t) < ξ i (t) = p(t) < p * for each t ∈ (t 0 , t 1 ]. By Claim 6, one at least of the two strategies σ i and σ j -say, σ 2 -, assigns probability zero to t 1 . This implies that γ 1 (t 1 , σ 2 ) is equal to the equilibrium payoff γ 1 (σ 1 , σ 2 ), in contradiction with O4. This implies S i = S j , as claimed. We denote by S the common set.

Next, using both Claim 6, O2 and O3, it is easy to check that T p = min S. If S failed to be an interval, there would exist t 0 , t 1 ∈ S, with T p ≤ t 0 < t 1 and (t 0 , t 1 )∩S = ∅. We now essentially repeat the previous paragraph. By Claim 6, there is a player, say j, such that σ j ({t 1 }) = 0, which implies γ i A (t 1 , σ j ) = γ i A (σ i , σ j ), in contradiction with O4.

In order to complete the discussion of Theorem 4, we argue informally that there is at most one equilibrium. (Existence is proven in the Appendix.) Let as above an equilibrium (σ 1 , σ 2 ) be given. Using Claim 7, we denote by [T p , T ] the common support of the two distributions σ 1 and σ 2 . Wlog, we may further assume that σ 2 is a non-atomic distribution. This implies that the indifference condition γ 1 A (t, σ 2 ) = γ 1 A (σ 1 , σ 2 ) holds for each t ∈ [T p , T ]. We will first show that this indifference condition uniquely pins down both the endpoint T and the distribution σ 2 . We next claim that σ 1 has to be equal to σ 2 , thereby establishing uniqueness.

We describe the distribution σ 2 by its cdf,

F 2 (t) = σ 2 ([0, t]) = σ 2 ([T p , t]) (t ∈ [T p , T ]). Note that F 2 (t) is also equal to P(θ 2 ≤ t | R 2 = B)
, and that the conditional probability

F2 (t) := P(θ 2 ≤ t | R 2 = G) is related to F 2 through the identity 1 -F2 (t) = P(θ 2 ≤ t, ≤ τ 2 | R 2 = G) + P(θ 2 ≤ τ 2 ≤| R 2 = G) = e -λt (1 -F 2 (t)) + t 0 λe -λt (1 -F 2 (z))dz.
We place ourselves at some time t ∈ [T p , T ], and will write an expansion of the equality

γ 1 A (t + dt, σ 2 ) = γ 1 A (t, σ 2 )
, as dt → 0. If either τ 1 < t or θ 2 < t, the outcomes induced by the two profiles (t, σ 2 ) and (t+dt, σ 2 ) are the same. We henceforth condition on the event min(τ 1 , θ 2 ) ≥ t.

Consider the three events

A := {τ 1 < t + dt}, B := {τ 1 ≥ t + dt, θ 2 ∈ [t, t + dt)}, and
C := {τ 1 ≥ t + dt, θ 2 ≥ t + dt}. (Conditional on min(τ 1 , θ 2 ) ≥ t), the probability of A is of the order of λξ 1 (t)dt, while the probability of B is P(θ 2 ∈ [t, t + dt) | min(τ 1 , θ 2 ) ≥ t).
The continuation payoff induced by (t + dt, σ 2 ) as of time t, is approximately equal γ on the event A, equal to W (φ(t)) on the event B, and is exactly equal to se -rdt on the event C, while the continuation payoff induced by (t, σ 2 ) is equal to s in all three cases.

Thus, the equality

γ 1 A (t + dt, σ 2 ) = γ 1 A (t, σ 2 ) writes (γ -s)λξ 1 (t) + (W (φ(t)) -s) P(θ 2 ∈ [t, t + dt) | min(τ 1 , θ 2 ) ≥ t) dt = rs.
This 'shows' that the limit µ 1 (t) := lim dt→0

P(θ 2 ∈[t,t+dt)|min(τ 1 ,θ 2 )≥t) dt
is well-defined and solves (γs)λξ 1 (t) + (W (φ(t))s)µ 1 (t) = rs.

(5)

The value of the limit µ 1 (t) may be interpreted as a conditional hazard rate for θ 2 , conditional on τ 1 ≥ t. Note that the probability P(τ 1 ≥ t, θ 2 ∈ [t, t + dt)) is given by

q GG e -λt + q GB ( F2 (t + dt) -F2 (t)) + q GB e -λt + q BB (F 2 (t + dt) -F 2 (t)),
where q ω 1 ω 2 stands for the prior probability that R 1 = ω 1 and R 2 = ω 2 ). A similar formula holds for P(τ 1 ≥ t, θ 2 ≥ t). Letting dt → 0 and plugging into (5), standard algebraic manipulations then show that F 2 solves the integro-differential equation ( 1) on the interval [T p , T ].

Since σ 2 is concentrated on [T p , T ], one has moreover F 2 (T p ) = 0 and

Tp 0 e -λx (1 -F (x)) = 1 -e -λTp λ
. There is a unique solution, say F2 , to the equation ( 1) on [T p , T φ ], that satisfies these initial conditions. Therefore, it must be that T := min{t : F2 (t) = 1}, and F 2 coincides with F2 on the interval [T p , T ]. This shows the uniqueness of T and of F 2 , as desired.

We turn to the strategy σ 1 . Since σ 1 has no atom in the interval (T p , T ], one has γ 2 A (t, σ 1 ) = γ 2 A (t ′ , σ 1 ) for every t, t ′ ∈ (T p , T ], so that the hazard rate µ 2 of player 1's exit decision is related to player 2's belief ξ 2 through the equation ( 5), for every t ∈ (T p , T ]. The economic intuition behind the uniqueness claim is as follows.

Assume for a moment that σ 1 has an atom located at T p : σ 1 ({T p } > 0. Then, when at time T p + (if player 1 remains active), player 2 is more pessimistic than player 1: ξ 2 (T p +) < ξ 1 (T p +). Since both players 1 and 2 are indifferent between dropping out immediately, and waiting a further infinitesimal amount of time, it must be that player 2 assigns higher chances than player 1 does, to the fact that he will receive good news in this time interval. In other words, the probability of player 1 dropping out should be higher than the probability of player 2 dropping out, to compensate for the fact that player 2 is more pessimistic. Indeed, from ( 5), one has µ 2 (t) > µ 1 (t) as soon as ξ 1 (t) > ξ 2 (t). Should players remain active, this will make player 2 even more pessimistic than player 1. Proceeding "inductively", one obtains more generally that F 1 (t) > F 2 (t) for all t ∈ [T p , T ] and therefore, F 1 (t) = 1 for some t < T -in contradiction with the fact that σ 1 and σ 2 have the same support.

The case of bad news

We now turn to the case where the correlation between the two risky arms is positive. Again, we derive direct implications of the observations O1-4 and of Claims 5 and 6.

We first check that the main qualitative insight is valid (Proposition 2). Let (σ 1 , σ 2 ) be an equilibrium and assume by way of contradiction that ξ i (t) < p * for some t ∈ S i . Since ρ > 0, the two beliefs φ(t) and ξ i (t) satisfy φ(t) ≤ ξ i (t). Using (4), this yields γ i A (t+, σ j ) = γ i A (t, σ j ). This implies in turn that the payoff γ i A (t, σ j ) induced by the strategy t must be equal to the equilibrium payoff γ i A (σ i , σ j ), in contradiction with O3. Thus, whenever player i is willing to drop out, his belief ξ i is equal to the one-player threshold, p * .

Uniqueness of the symmetric equilibrium

Let (σ, σ) be a symmetric equilibrium, with support S. Using Claim 6, the distribution σ is necessarily non-atomic. By the Proposition 2 just proven, one must have ξ i (t) = p * for each t ∈ S. The only way for a player's belief to remain constant over time is when the bad news coming from one's own arm is exactly offset by the good news coming from the other player, at any point in time. This suggests that the support S of σ must be an interval, say [T 1 , T 2 ].

Note now that ξ i (T 1 ) = p(T 1 ): indeed, since player j never stops prior to T 1 , the event θ j ≥ T 1 is uninformative. On the other hand, player j stops for sure before T 2 , unless he has hit a success: thus, ξ i (T 2 ) = ψ(T 2 ). Since ξ i (T 1 ) = ξ i (T 2 ) = p * , T 1 and T 2 must therefore be equal to T p and T ψ respectively. Finally, the distribution σ is uniquely defined by the condition that ξ i (t) = p * for each t ∈ [T p , T ψ ). While this establishes uniqueness, this does not prove existence of a symmetric equilibrium. The existence proof is common to the case where ρ < 0, and is postponed to the Appendix.

Pure equilibria

In contrast with the case where ρ > 0, there may be many equilibria with finite support, as stated in Proposition 3. Let first (t 1 , t 2 ) be a tentative pure equilibrium. By Claim 6, one has t 1 = t 2 . Assume wlog that t 1 < t 2 . As above, one then has ξ 1 (t 1 ) = p(t 1 ) and ξ 2 (t 2 ) = ψ(t 2 ) and therefore, t 1 = T p and t 2 = T ψ . We now argue that (T p , T ψ ) is indeed a pure equilibrium.

When facing T p , the payoff function t → γ 2 A (t, σ 1 ) of player 2 is increasing over [0, T ψ ] since ξ 2 (t+) > p * for every t < T ψ , and decreasing beyond T ψ , with a discontinuity at T p . Therefore, T ψ is the unique best reply of player 2 to T p .

When facing T ψ , and for t ≤ T ψ , the payoff of player 1 is the same as in the decision problem P, since player 1 does not deduce any information from the behavior of player 2 up to time t. This payoff is therefore increasing over [0, T p ], and decreasing on [T p , T ψ ]. Should player 1 choose to wait beyond T ψ , and to learn the decision of player 2 at time T ψ , his belief would then jump either to φ(T ψ ) or to ψ(T ψ ), depending on that decision. But since φ(T ψ ) < ψ(T ψ ) = p * , player 1 will then find it optimal to drop out anyway. Hence T p is the unique best reply to the strategy T ψ .

Aymmetric equilibria

The proof that equilibria may exist with a finite support of arbitrary size is a bit delicate and we briefly discuss the main ideas. Let a tentative equilibrium (σ 1 , σ 2 ) be given, where σ i assigns positive probability to exactly k dates. By Claim 6, the two equilibrium supports S 1 and S 2 are disjoint. Since ξ i (t i ) = p * , it must be that the two sets S 1 and S 2 are interlaced : any two consecutive dates in S i are separated by exactly one date in S j . The equilibrium logic is as follows. If player i does not drop out at some t i ∈ S i , this is good news for player j, whose belief jumps above p * , and then declines continuously until it reaches p * at the next date t j in S j . Meanwhile, player i's belief declines below p * until date t j . If player j does not drop out at that date, player i' s belief jumps above p * , and then declines continuously until it reaches p * at the next date in S i . And so on.

We illustrate the proof in the specific case where k = 2, and S i = {t i , ti }. Assuming wlog t 1 < t 2 , the same arguments as above imply t 1 = T p and t2 = T ψ . We are thus left with four unknownst 2 and t1 , and the weight assigned by σ i to t i -, and four equilibrium conditions: the belief conditions ξ 1 ( t1 ) = ξ 2 (t 2 ) = p * and the indifference condition for player i at time t i . Existence is proven using a fixed point argument in the convex and compact set of pairs (t 2 , t1 ) such that T p ≤ t 2 ≤ t1 ≤ T ψ . In a nutshell, the proof goes as follows. Given t1 ∈ [T p , T ψ ], we prove that there is a unique t 2 ∈ [T p , t1 ] and a unique weight on t 2 such that (i) the belief held by player 1 at t1 is equal to t * and (ii) the payoffs induced by the two strategies T p and t1 are the same. Moreover, t 2 is continuous w.r.t. t1 and t 2 ∈ (T p , t1 ) whenever t1 > T p . Using a similar argument, we prove that for given t 2 , there is a unique t1 ∈ [t 2 , T ψ ] and a unique weight on t 1 = T p such that (i) the belief of player 2 at t 2 is p * and (ii) the payoffs induced by the two Claim 9 The function π is of class C 1 and its derivative is given by

π′ (t) = e -rt P(min(τ 1 , τ 2 ) ≥ t) × {λφ(t) (W (ψ(t)) + γ -2s) -rs} , t ∈ R + . (9)
Proof. Fix t 0 ∈ R + . We will prove the claim relative to the right-derivative. Similar arguments apply for the existence of a left derivative.

Observe first that, for t > t 0 ,

P(τ 1 , τ 2 ∈ [t 0 , t]) ≤ P(τ 1 , τ 2 ∈ [t 0 , t] | R 1 = R 2 = G) = e -2λt 1 -e -λ(t-t 0 ) 2 ≤ λ 2 (t -t 0 ) 2 .
Therefore, lim

tցt 0 P(τ 1 , τ 2 ∈ [t 0 , t]) t -t 0 = 0. (10)
Thus, when taking the limit of π(t)π(t 0 ) tt 0 (see ( 6)), the indicators 1 t 0 ≤τ 1 <min(τ 2 ,t) and 1 t 0 ≤τ 2 <min(τ 1 ,t) which appear in the first two expectations can be replaced with 1 t 0 ≤τ 1 <t and 1 t 0 ≤τ 2 <t respectively.

Next, consider each of the (simplified) two expectations in turn (divided by

P(min(τ 1 , τ 2 ) ≥ t 0 )). Observe first that 1 t -t 0 E e -r(τ 1 -t 0 ) W (ψ(τ 1 ))1 τ 1 ∈[t 0 ,t) | min(τ 1 , τ 2 ≥ t 0 ) = φ(t 0 ) 1 t -t 0 t t 0
e -r(x-t 0 ) W (ψ(x))λe -λ(x-t 0 ) dx, which converges to λφ(t 0 )W (ψ(t 0 )) when t ց t 0 , since W (•) and ψ(•) are continuous.

Observe next that 1 tt 0 E e -r(τ 2 -t 0 1 τ 2 ∈[t 0 ,t) | min(τ 1 , τ 2 ≥ t 0 converges to λφ(t 0 ), for similar reasons.

We finally deal with the last term in the expression of π(t)π(t 0 ) in (6). One has

1 P(min(τ 1 , τ 2 ) ≥ t 0 ) × 1 t -t 0 e -r(t-t 0 ) P (min(τ 1 , τ 2 ) ≥ t) -P(min(τ 1 , τ 2 ) ≥ t 0 ) = 1 t -t 0 e -r(t-t 0 ) P(min(τ 1 , τ 2 ) ≥ t | min(τ 1 , τ 2 ) ≥ t 0 ) -1 = 1 t -t 0 (e -r(t-t 0 ) -1)P (min(τ 1 , τ 2 ) ≥ t | min(τ 1 , τ 2 ) ≥ t 0 ) -P (min(τ 1 , τ 2 ) < t | min(τ 1 , τ 2 ) ≥ t 0 ) .
The first term converges to -r. Using (10), the second term has the same limit, namely -2λφ(t 0 ), as the expression

-(P(τ 1 ∈ [t 0 , t) | min(τ 1 , τ 2 ) ≥ t 0 ) + P(τ 2 ∈ [t 0 , t) | min(τ 1 , τ 2 ) ≥ t 0 )) .
The result follows when adding these different limits.

We now conclude the proof of Proposition 1. Let T * * ∈ R + be any point at which π reaches its maximum. One has π′ (T * * ) = 0, that is

λφ(T * * )W (ψ(T * * )) + λφ(T * * )γ -s(r + 2λφ(T * * )) = 0 or equivalently, λφ(T * * ) (W (ψ(T * * ) + γ -2s) = rs. ( 11
)
Since φ is decreasing, positive and continuous, and since W (ψ(•)) is non-increasing, Equation ( 11) has a unique solution.

Recall now that T φ solves λφ(T φ )(γs) = rs.

Since W (ψ(T * * )) ≥ s, this implies φ(T * * ) ≤ φ(T φ ), hence T * * ≥ T φ . If ρ < 0, one has ψ(T * * ) ≤ φ(T * * ) ≤ φ(T φ ) = p * , hence W (ψ(T * * )) = s, so that Equation (3) reduces in that case to λφ(T * * )(γ -s) = rs, and T * * = T φ .
If instead ρ > 0, then ψ(T * ) > φ(T φ ), hence W (ψ(T φ )) > s, so that T φ is not a solution to (11). In that case, one therefore has T * * > T φ .

B The actions scenario

In this section, we provide the proofs of all the results relative to the actions scenario. The section is organized as follows. We first introduce in Section B.1 some notation and prove a few properties, which are valid irrespective of the sign of the correlation. Next, we prove that the marginal value of the informational externality is zero when ρ > 0 (Proposition 1 in the paper), and characterize the unique symmetric equilibrium in that case (Theorem 1 in the paper). We then prove the existence and the uniqueness of the equilibrium when ρ < 0 (Theorem 4 in the paper), and provide a characterization of the equilibrium. Finally, we come back to the case of a positive correlation, and prove the multiplicity of asymmetric equilibria (Proposition 2 in the paper).

B.1 Technical Preliminaries

We let a strategy σ j of player j be fixed throughout this section. Recall that the belief function ξ i is defined by

ξ i (t) = P(R i = G | τ i ≥ t, θ j ≥ t) (t ∈ R + )
, which is computed under the assumption that player j is using the strategy σ j . It is readily checked that ξ i (•) is leftcontinuous over R + , and has a right-limit at each t ∈ R + , which is given by ξ i (t+) := P(R i = G | τ i > t, θ j > t). In addition, ξ i is continuous at t ∈ R + if and only if P(θ j = t) = 0 that is, if and only if σ j ({t}) = 0.

B.1.1 Expected Payoffs

This section serves as a reminder for some notation, and provides a few elementary properties of the payoff function γ A .

When facing the strategy σ j , and when using the strategy t, the continuation payoff of player i at time min(t, τ i , θ j ) is equal to (i) s if θ j ≥ t and τ i ≥ t, (ii) γ if τ i ≤ θ j and τ i < t, and (iii)

W (φ(θ j )) if θ j < min(t, τ i ).
Since τ i has a density conditional on R i = G, one has P(τ i = θ j < +∞) = 0, and player i's expected payoff under (t, σ j ) is thus given by

γ i A (t, σ j ) = se -rt P(θ j ≥ t, τ i ≥ t) + γE e -rτ i 1 τ i <min(θ j ,t) + E e -rθ j W (φ(θ j ))1 θ j <min(τ i ,t) .
In particular, for t 0 < t, one has

γ i A (t, σ j ) -γ i A (t 0 , σ j ) = se -rt P(θ j ≥ t, τ i ≥ t) -se -rt 0 P(θ j ≥ t 0 , τ i ≥ t 0 ) (12)
+γE e -rτ i 1 t 0 ≤τ i <min(θ j ,t) + E e -rθ j W (φ(θ j ))1 t 0 ≤θ j <min(τ i ,t) . (

By dominated convergence, the map t → γ i A (t, σ j ) is left-continuous over R + , and admits a right-limit at each t ∈ R + , which we denote γ i A (t+, σ j ), and which is given by

γ i A (t+, σ j ) -γ i A (t, σ j ) = e -rt P(θ j = t, τ i ≥ t) × (W (φ(t)) -s) . (14)
Since W (p) ≥ s for each p ∈ [0, 1], this implies that γ i A (t+, σ j ) ≥ γ i A (t, σ j ), and that equality holds when either σ j ({t}) = 0, or φ(t) ≤ p * .

If σ i is a best reply of player i to σ j , then γ i A (t, σ j ) = γ i A (σ i , σ j ) for σ i -a.e. t ∈ [0, +∞]. In particular, for a given t ∈ R + , one has γ i A (t, σ j ) = γ i A (σ i , σ j ) whenever σ j ({t}) = 0, and t belongs to the support of σ i . We will also rely on the following fact. Fact 0. For every t in the support of σ i , one has

γ i A (t+, σ j ) = γ i A (σ i , σ j ).
Indeed, note that any such t is a limit of a sequence (t n ) such that 18 γ i A (t n , σ j ) = γ i A (σ i , σ j ). Note next that lim sup n→+∞ γ i A (t n , σ j ) ≤ γ i A (t+, σ j ) for any sequence t n → t. This yields Fact 0.

It will be convenient to study and characterize equilibria by means of the differential properties 18 Otherwise indeed, one would have γ i A (x, σ j ) < γ i A (σ i , σ j ) for all x in some neighborhood of t, and player i would be better off assigning probability zero to that neighborhood. of t → γ i A (t, σ j ). We here gather a few simple observations, to which we will repeatedly refer later. All these observations are valid, irrespective of the strategy σ j .

We let x > 0 be given, and will let y > x vary. 19 Since τ i follows an exponential distribution if R i = G, one has:

Fact 1. lim yցx 1 y-x P(τ i < y | min(θ j , τ i ) ≥ x) = λξ i (x).
Since P(θ j ∈ (x, y)) converges to zero as y → x, and since θ j and τ i are conditionally independent given the types (R i , R j ), Fact 1 implies that

lim yցx 1 y -x P(x < θ j , τ i ≤ y) = 0. ( 15
)
Next, observe that e -r(y-x) P(min(θ j , τ i ) ≥ y | min(θ j , τ i ) ≥ x) -1 is equal to (e -r(y-x) -1)P(min(θ j , τ i ) ≥ y | min(θ j , τ i ) ≥ x) -P(min(θ j , τ i ) < y | min(θ j , τ i ) ≥ x).

Combining (15) with Fact 1, this easily yields Fact 2 below.

Fact 2. The sum of 1 yx e -r(y-x) P(min(θ j , τ i ) ≥ y | min(θ j , τ j ) ≥ x) -1 and of 1 yx P(θ j < y | min(θ j , τ j ) ≥ x) converges to r + λξ i (x) as y ց x.

Let β(•) be a C 1 function of time. For y > x close to x, the difference sup t∈[x,y] |β(t)β(x)| is at most of the order of (yx)β ′ (x). Since lim y→x P(θ j ∈ (x, y)) = 0, and together with the previous fact, this implies Fact 3 below.

Fact 3. Let β be a C 1 function. The difference between 1 y -x E[β(θ j )1 θ j <min(y,τ i ) | min(τ i , θ j ) ≥
x] and 1 yx β(x)P(θ j < y | min(τ i , θ j ) ≥ x) converges to zero as y → x.

For similar reasons, one has Fact 3b below.

Fact 3b. Let β be a C 1 function. The difference between 1 y -x E[β(τ i )1 τ i <min(y,θ j ) | min(τ i , θ j ) ≥ x] and 1 y -x E[β(τ i )1 τ i <y | min(τ i , θ j ) ≥ x)
converges to zero as y ց x. In addition,

lim yցx 1 y-x E[β(τ i )1 τ i <min(y,θ j ) | min(τ i , θ j ) ≥ x] = λξ i (x)β(x).
19 We assume without further notice that P(θ j > x) > 0.

B.1.2 The actions game and the decision problem P

We here discuss the relation between optimal policies in P, and monotonicity properties of the map t → γ i A (t, σ j ). Let first I be an interval that is assigned probability zero by σ j . Then, while in the time interval I, player i is 'alone' and the comparison between two policies t and t in I is driven by the map W . Formally, let t 0 , t, t ∈ I be such that t 0 < t < t. By (12), one has

γ i A (t, σ j ) -γ i A (t, σ j ) e -rt P(τ i ≥ t, θ j ≥ t 0 ) = se -r(t-t) P (τ i ≥ t | τ i ≥ t, θ j ≥ t 0 )+γE e -r(τ i -t) 1 τ i <t | τ i ≥ t, θ j ≥ t 0 -s.
(16) The right-hand side of equation ( 16) is equal to the payoff induced in the decision problem P by the policy tt, when starting with a belief equal to ξ i (t). This immediately implies the following fact.

Fact 4. Under the above assumptions, one has γ i A (t, σ j ) < γ i A (t, σ j ) if ξ i (t) < p * , and

γ i A (t, σ j ) > γ i A (t, σ j ) if ξ i (t) > p * . 20
Our next observation has a straightforward intuition, but the formal proof requires a few details.

Fact 5. Assume that ξ i (t 0 +) > p * . Then γ i A (t, σ j ) > γ i A (t 0 , σ j ) for all t > t 0 close enough to t 0 .

The intuition is as follows. We place ourselves at time t 0 , in the event where τ i ≥ t 0 and θ j ≥ t 0 . Player i may choose to exit (strategy t 0 ), and thereby to get a continuation payoff equal to s. Or he may choose to wait for player j's decision, and then play optimally. Since ξ i (t 0 +) > p * , the latter choice yields a higher payoff.

The formal proof goes as follows. Since ξ i (t 0 +) > p * , the derivative of the map W (•), evaluated at ξ i (t 0 +), is positive. We will prove that lim inf

tցt 0 γ i A (t, σ j ) -γ i A (t 0 , σ j ) t -t 0 ≥ e -rt 0 W ′ (ξ i (t 0 +))P(τ i ≥ t 0 , θ j > t 0 ),
which yields the result. Since W (φ(x)) ≥ s for every x ≥ t 0 , one has using (12), γ i A (t, σ j )γ i A (t 0 , σ j ) ≥ se -rt P (τ i ≥ t, θ j ≥ t)se -rt 0 P (τ i ≥ t 0 , θ j ≥ t 0 ) +sE e -rθ j 1 t 0 ≤θ j <min(τ i ,t) + γE e -rτ i 1 t 0 ≤τ i <min(θ j ,t) .

When moving the event θ j = t 0 from the third to the second term, and using the fact that τ i has a density, one obtains

γ i A (t, σ j ) -γ i A (t 0 , σ j ) ≥ se -rt P(τ i ≥ t, θ j ≥ t) -se -rt 0 P(τ i ≥ t 0 , θ j > t 0 ) (17)
+sE e -rθ j 1 t 0 <θ j <min(τ i ,t) + γE e -rτ i 1 t 0 ≤τ i <min(θ j ,t) .

≥ se -rt P(τ i ≥ t, θ j > t 0 )se -rt 0 P(τ i ≥ t 0 , θ j > t 0 ) + γE e -rτ i 1 t 0 ≤τ i <min(θ j ,t)

Using Fact 3b, this implies that 1 e -rt 0 P(τ i ≥ t 0 , θ j > t 0 ) lim inf

tցt 0 γ i A (t, σ j ) -γ i A (t 0 , σ j ) t -t 0 is at least equal to lim tցt 0 1 t -t 0 se -r(t-t 0 ) P(τ i ≥ t | τ i ≥ t 0 , θ j > t 0 ) + γE e -r(τ i -t 0 ) 1 τ i <t | τ i ≥ t 0 , θ j > t 0 -s ,
which is equal to W ′ (ξ i (t 0 +)). This concludes the proof of Fact 5.

Since the inequality ξ i (t 0 +) > p * implies that ξ i (t+) > p * for all t > t 0 close enough to t 0 , the next fact is a direct consequence of Fact 5. Fact 6 Assume that ξ i (t 0 +) > p * . Then t → γ i A (t, σ j ) is increasing over [t 0 , t 0 + ε], for ε > 0 small enough.

B.2 A priori best-reply properties

In this section, we let σ j be an arbitrary strategy of player j, and we let σ i be a best reply to σ j . We denote by S i and S j the supports of σ i and σ j .

We derive two properties which will help us pinning down the equilibrium set. In Lemmas 1 and 2, we first give a formal content to the intuitive assertion that a player will not drop as long as his belief exceeds p * . Since the belief ξ i need not be continuous, this will surprisingly turn out to be somewhat delicate. Next, in Lemma 3, we prove that if there is there is a positive probability that player j will drop at t 0 , then it cannot be optimal for player i to drop out just before t 0 .

Lemma 1 One has

• ξ i (t+) ≤ p * for all t ∈ S i . • ξ i (t)(= ξ i (t+)) = p * for all t ∈ S i \ S j .
Proof. We start with the first statement. Recall that, by the best-reply property, one has γ i A (t, σ j ) = γ i A (σ i , σ j ) for σ i -a.e. t ∈ S i .

We argue by contradiction, and assume that ξ i (t 0 +) > p * for some t 0 ∈ S i . By Fact 6, there is t > t 0 such that γ i A ( t, σ j ) > γ i A (t, σ j ) for all t ∈ [t 0 , t), hence σ i ([t 0 , t)) = 0. On the other hand, since γ i A (t 0 +, σ j ) ≥ γ i A (t 0 , σ j ) and since γ i A (•, σ j ) is left-continuous, one also has γ i A (t, σ j ) < γ i A ( t, σ j ) for all t ∈ [t, t 0 ], as soon as t < t 0 is close enough to t 0 . Therefore, σ i ([t, t 0 ]) = 0, and thus σ i ([t, t)) = 0. But this contradicts the assumption that t 0 ∈ S i .

We turn to the second statement. Let I be an open interval containing t, and such that I ∩ S j = ∅. In particular, x → γ i A (x, σ j ) is continuous on I. Thus,

γ i A (x, σ j ) ≤ γ i A (t, σ j ) = γ i A (σ i , σ j ),
for each x ∈ I. The result is then a direct consequence of Fact 4.

Lemma 2 Let t 0 ∈ R + . If ξ i (t 0 ) > p * , then σ i ([t 0ε, t 0 ]) = 0 for each ε > 0 small enough.

Proof. Since ξ i is left-continuous, one has ξ i (t) > p * for all t ∈ [t, t 0 ], as soon as t < t 0 is close enough to t 0 . By Fact 6, this implies that t → γ i A (t, σ j ) is increasing on the interval [t, t 0 ]. Hence, σ i ([t, t 0 )) = 0.

We next discuss two cases. If σ j ({t 0 }) = 0, then ξ i (t 0 +) = ξ i (t 0 ) > p * . Using the proof of Fact 6, the map t → γ i A (t, σ j ) is increasing on the interval [t 0 , t], as soon as t > t 0 is close enough to t 0 . It follows in that case that σ i ([t 0 , t)) = 0, as desired.

If instead σ j ({t 0 }) > 0, then ξ i (•) is discontinuous at t 0 . Since ξ i (t 0 ) is a weighted average of φ(t 0 ) and of ξ i (t 0 +), one has either φ(t 0 ) > p * or ξ i (t 0 +) > p * . In the former case, one has W (φ(t 0 )) > s, and thus γ i A (t 0 +, σ j ) > γ i A (t 0 , σ j ), using (14). It follows that σ i ({t 0 }) = 0. In the latter case, by Fact 6 and as in the preceding paragraph, one has σ i ([t 0 , t)) = 0 for t > t 0 close enough to t 0 .

We stress that Lemmas 1 and 2 do not rule out the possibility that ξ i (t) > p * for some t ∈ S i . Indeed, Lemmas 1 and 2 are consistent with a situation in which ξ i (t 0 ) > p * ≥ ξ i (t 0 +), and a strategy σ i that assigns probability zero to the interval [t, t 0 ], yet assigns a positive probability to the interval (t 0 , t 0 + ε), for all ε > 0.

While the conclusion that ξ i (t) ≤ p * for all t ∈ S i is nevertheless valid, it will only be established as a corollary to our characterization results, and we do not know of any simple and direct proof.

Lemma 3 Let t 0 ∈ R + be given. Assume that σ j ({t 0 }) > 0. Then σ i ([t 0ε, t 0 ]) = 0 for some ε > 0.

Proof. We handle first a few easy cases. If γ i A (t 0 +, σ j ) > γ i A (t 0 , σ j ), then by left-continuity, one has γ i A (t, σ j ) < γ i A ( t, σ j ) for all t, t close enough to t 0 and such that t ≤ t 0 < t. The result follows in that case.

If now ξ i (t 0 ) > p * , the conclusion follows by Lemma 2. We are thus left with the case in which ξ i (t 0 ) ≤ p * and γ i A (t 0 +, σ j ) = γ i A (t 0 , σ j ). Using equation ( 14), one must have W (φ(t 0 )) = s, so that φ(t 0 ) ≤ p * . By Lemma 1, one has ξ i (t 0 +) ≤ p * as well. Since ξ i is discontinuous at t 0 , and since ξ i (t 0 ) is a weighted average of ξ i (t 0 +) and of φ(t 0 ), this implies that ξ i (t 0 ) < p * .

We claim that one must then have lim sup

tրt 0 γ i A (t 0 , σ j ) -γ i A (t, σ j ) t 0 -t < 0. ( 20
) Since γ i A (t 0 , σ j ) = γ i A (t 0 +, σ j ) = γ i A (σ i , σ j )
, the conclusion will follow. We rely on equation (12). Since t → W (φ(t)) is a C 1 function, with W (φ(t 0 )) = s, and by Facts 3 and 3b, the difference between 1 e -rt P(τ

i ≥ t, θ j ≥ t) × γ i A (t 0 , σ j ) -γ i A (t, σ j ) t 0 -t and 1 t 0 -t se -r(t 0 -t) P(τ i ≥ t 0 | τ i ≥ t, θ j ≥ t) + γE e -r(τ i -t) 1 t≤τ i <t 0 | τ i ≥ t, θ j ≥ t -s (21)
converges to zero. Observe that the expression in ( 21) is equal to the incremental payoff over s (normalized by t 0t) in P when using the policy t 0t, and when starting from a belief of ξ i (t). Since ξ i (t 0 ) < p * , the limit of this expression is therefore negative. This proves (20).

B.3 On non-atomic equilibria

Again, we let a strategy σ j of player j be given. We set F j (t) := σ j ([0, t]), which is also equal to P(θ j ≤ t | R j = B). We also introduce H j (t) := t 0 e -λx (1 -F j (x))dx. In this section, we prove the next result.

Proposition 5 Let I be an interval in R + with non-empty interior, such that σ j ({t}) = 0 for each t ∈ I. Then the following two statements are equivalent:

• On the interval I, the map t → γ i A (t, σ j ) is constant;

• On the interval I, the map H j is C 2 and is a solution to the linear, second-order differential equation

W (φ(t)) -s λ(γ -s) H ′′ (t) + λH ′ (t) = (φ(t) -p * )H ′ (t) + λφ(t)(ψ(t) -p * )H(t). (22)
When plugging (25), ( 26) and ( 27) into (24), when using the relation between Fj and F j , and after obvious algebraic manipulations, one obtains (W (φ(x))s) q GG e -2λx + 2q GB e -λx + q BB F ′ j (x)

= (1 -F j (x)) rs q GG e -2λx + 2q GB e -λx + q BBλ(γs)(q GG e -2λx + q GB e -λx ) (29)

+ x 0 λe -λt (1 -F j (t))dt × rs(q GG e -λx + q GB ) -λ(γ -s)q GG e -λx . (30)
Observe next that q GG e -2λx + 2q GB e -λx + q BB = P(τ i ≥ x, τ j ≥ x), while

q GG e -2λx + q GB e -λx = P(R i = G, τ i ≥ x, τ j ≥ x) = φ(x)P(τ i ≥ x, τ j ≥ x).
On the other hand,

q GG e -λx q GG e -λx + q GB = P(R i = G, R j = G, τ i ≥ x) P(R j = G, τ i ≥ x) = ψ(x).
Therefore, one has

(W (φ(x)) -s) F ′ j (x) = (1 -F j (x)) (rs -λ(γ -s)φ(x)) +φ(x)e λx (rs -λ(γ -s)ψ(x)) x 0 λe -λt (1 -F j (t))dt
or equivalently, using p * := rs λ(γ-s) ,

W (φ(x)) -s λ(γ -s) F ′ j (x) = (1 -F j (x))(p * -φ(x)) + λφ(x)(p * -ψ(x))e λx x 0 e -λt (1 -F j (t))dt. Since H j (x) = t 0 e -λt (1 -F j (t))dt, it follows that H j solves the equation W (φ(x)) -s λ(γ -s) H ′′ j (t) + λH ′ j (t) = (φ(x) -p * )H ′ j (t) + λφ(x)(ψ(x) -p * )H j (t)dt
on the interval I, as desired.

Conversely, if H j (x) :=

x 0 e -λt (1 -F j (t)) is a C 2 function and solves (23) on the interval, then the above computation shows that the map t → γ i A (t, σ j ) is differentiable on I, with a derivative equal to zero throughout I. It follows that it is a constant map, as desired.

Proof. Since (σ, σ) is an equilibrium and by Lemma 3, the distribution σ has no atom, and thus ξ 1 (•) = ξ 2 (•) is continuous on S. By Proposition 3, ξ 1 = ξ 2 is equal to p * on S. Finally, if I is any interval such that σ(I) = 0, then ξ i is decreasing on I. This implies that S is an interval, say [T 1 , T 2 ]. Since σ([0, T 1 ]) = 0, and σ([0, T 2 ]) = 1, one has respectively ξ i (T 1 ) = p(T 1 ) and ξ i (T 2 ) = ψ(T 2 ), hence T 1 = T p , and T 2 = T 22) boils down to the first-order equation

H ′ (x) = λφ(x)(ψ(x) -p * ) p * -φ(x) H(x). (31)
This equation has a unique solution such that H(T p ) = 1e -λTp λ . This shows that a symmetric equilibrium, if it exists, must be unique. . We now prove that F is the cumulative of some mixed strategy, which forms a symmetric equilibrium. Observe first that, using (31) and the boundary condition, one has F (T p ) = 0, and F (T ψ ) = 1. It can be checked that the derivative of F is of the sign of ρ on the interval [T p , T ψ ], so that F is increasing. 21 Let σ be the probability distribution defined by σ([0, t]) = F (t). Note that σ has no atom. When facing the strategy σ, the belief of player i solves ξ i (t) = P(R i = G | τ i ≥ t) > p * for t < T p , and ξ i (t) = ψ(t) < p * for t > T ψ . Hence any best-reply of player i to σ assigns probability 1 to [T p , T ψ ]. By Proposition 5, the map t → γ i A (t, σ) is constant on [T p , T ψ ]. Hence, any strategy that assigns probability 1 to [T p , T ψ ] is a best-reply to σ. Therefore, (σ, σ) is an equilibrium.

B.5 Proof of Proposition 3

We assume here that the two arms are negatively correlated, ρ < 0. We prove that the game has a unique equilibrium, which happens to be symmetric. We first prove uniqueness, then existence, of an equilibrium. Thus, assume first that the game has an equilibrium, say (σ 1 , σ 2 ). Note that since ρ < 0, the belief ξ i (•) is decreasing on R + . We denote by S i the support of the distribution σ i .

Lemma 5 One has S 1 = S 2 = [T p , T ], for some T such that T p < T < T φ f.

Proof. We first argue by way of contradiction that S 1 = S 2 , and thus assume that there exists t 0 ∈ S 1 \ S 2 . By Lemma 1, one has ξ i (t 0 ) = p * . By Fact 6, and since ξ 1 is decreasing, one has t 0 = min S 1 . Hence, ξ 2 is simply equal to P(R 2 = G | τ 2 ≥ t) for t < t 0 . This implies that ξ 2 (t) ≥ ξ 1 (t), and thus, ξ 2 (t) > p * , for each t < t 0 . By Fact 6, one must then have t 0 < min S 2 .

Set t := inf{t > t 0 : t ∈ S 1 ∪ S 2 }. Observe that ξ 1 (t) < p * for each t ∈ (t 0 , min S 2 ), hence t / ∈ S 1 by Lemma 1. This implies that t = min S 2 . We claim that t belongs to S 1 as well. If instead t / ∈ S 1 . then ξ 2 (t) = p * by Lemma 1. On the other hand, and since t = min S 2 , one has ξ 1 (t) ≥ ξ 2 (t) for t ≤ t, which yields ξ 2 (t) < p * since ξ 1 (t) < p * for each t > t 0 -a contradiction, and thus t ∈ S 1 ∩ S 2 , as claimed. Note that we have proven as well that ξ i (t) < p * for i = 1, 2. By Lemma 3, at least one of the two strategies σ 1 , σ 2 does not have an atom at t. Fix i ∈ {1, 2} such that σ i ({t}) = 0. Then, γ j A (•, σ i ) is continuous at t, so that γ j A (t, σ i ) = γ j A (σ i , σ j ). By Fact 4, the map t → γ i A (t, σ j ) is decreasing over the interval [t 0 , t]. This is in contradiction with the equality γ j A (t, σ i ) = γ j A (σ i , σ j ). This concludes the proof that S 1 = S 2 .

We next prove, again by way of contradiction, that the common set S 1 = S 2 is an interval. Assume instead that there are two dates t < t in S, such that (t, t) ∩ S = ∅.

Wlog, using Lemma 3, assume that σ 2 ({t}) = 0, so that γ 2 A (t, σ 1 ) = γ 2 A (σ 1 , σ 2 ). Since t ∈ S, one has σ 1 ({t}) = σ 2 ({t}) = 0,22 one has ξ 2 (t) ≤ p * . By Fact 4 again, this implies that the map t → γ 2 A (t, σ 1 ) is decreasing over the interval (t, t] -a contradiction.

By Lemma 3, none of the two distributions σ 1 and σ 2 can possibly have an atom in (T p , T ], and at most one of the two distributions has an atom at T p . Assume that, say, that σ 1 has no atom. We denote by F 1 the cdf of σ 1 , and introduce H 1 (t) = t 0 e -λx (1 -F 1 (x))dx. Since σ 1 is non-atomic, the map t → γ 2 A (t, σ 1 ) is constant on the interval [T p , T ]. By Proposition 5, H 1 is C 2 on the interval [T p , T ], and solves the linear equation ( 22) on this interval. Since W (φ(t)) > s on the interval [T p , T φ ), the coefficient of H ′′ 1 does not vanish on this interval, and there is a unique solution to (22) on the interval [T p , T φ ) that assumes given initial values for H 1 (T p ) and H ′ 1 (T p ). On the other hand, note that H 1 satisfies the boundary condition H 1 (T p ) = 1-e -λTp λ , H ′ 1 (T p ) = e -λTp . Hence, H 1 , and therefore F 1 is uniquely defined. This in turn defines uniquely T as the smallest t such that F 1 (t) = 1.

We next proceed to show that σ 1 = σ 2 . We denote by F 2 the cdf of σ 2 , and introduce the map H 2 (t) := t 0 e -λx (1 -F 2 (x))dx, for t > T p . We argue by contradiction and assume that F 2 (T p ) > 0(= F 1 (T p )). Since σ 2 has no atom in (T p , T ], one has γ 1 A (t, σ 2 ) = γ 1 A (σ 1 , σ 2 ) for each t ∈ (T p , T ], and therefore, H 2 solves (22) on the interval (T p , T ]. In addition, when setting We set Ω(t) := (1 -F 1 (t))H 2 (t) -(1 -F 2 (t))H 1 (t). Note that Ω(T p ) > 0, while Ω( T ) = 0 since F 1 ( T ) = F 2 ( T ) = 1.

Set T := min{t ∈ [T p , T ] : Ω(t) = 0}. Since Ω( T ) = 0, one has T ∈ [T p , T ] and Ω(T ) = 0. Since Ω(T p ) > 0, one has Ω(t) > 0 for each t ∈ [T p , T ). From ( 22), one has

F ′ i (
By Lemma 6, H ′ is decreasing and positive on [T p , T φ ], and is therefore bounded. Note also that H is positive, increasing and bounded. Since the map x → W (φ(x))s is C 1 and vanishes for t = T φ , the first integral in (33) has a finite limit when t → T φ , while the second one converges to -∞, which implies H ′ (t) → -∞, a contradiction. Define T := min{t ∈ [T p , T φ ] : H ′ (t) = 0}. Define F by F (t) = 0 for t ≤ T p , F (t) = 1-e λt H ′ (t) for t ∈ [T p , T ], and F (t) = 1 for t > T ]. The map F is increasing on [T p , T ] by Lemma 6, and is continuous over R + . It is therefore the cdf of a non-atomic measure σ, which is concentrated on [T p , T ]. We now argue that (σ, σ) is a symmetric equilibrium.

Since σ is non-atomic, the map t → γ i A (t, σ) is continuous over the compact set R + ∪ {+∞}, and a best-reply therefore exists. When facing σ, one has ξ i (t) > p * for t < T p , and ξ i (t) < p * for t > T , hence the support of any best-reply map is included in [T p , T ]. By Proposition 5, the map t → γ i A (t, σ) is constant on [T p , T ], and thus, any strategy with support in [T p , T ] is a best-reply to σ.

B.6 Proof of Proposition 3

The proof of Proposition 3 may be the more delicate, and we will remain on the informal side, to save much on notation. 23 We let K ∈ N be given, and will prove the existence of an equilibrium with a support of cardinal K. Set Ω := {(u 1 , v 1 , . . . , u K , v K ) ∈ R 2K such that T p = u 1 ≤ v 1 ≤ • • • ≤ u K ≤ T ψ }. We will focus on strategies with a finite support S. Rather than

  ψ . Define F (t) := σ([0, t], and set H(t) := t 0 e -λx (1 -F (t))dt. By Lemma 4, H(T p ) = Tp 0 e -λt dt = 1e -λTp λ . Since σ has no atom, the payoff function t → γ i A (t, σ) is constant on the interval [T p , T ψ ]. Therefore, by Proposition 5, H is C 2 on the interval [T p , T ψ ] and solves the equation (22) on that interval. Since ρ > 0, one has φ(t) < p * , and therefore W (φ(t)) = s, for each t ∈ [T p , T ψ ]. Hence the equation (

  Conversely, consider the (unique) solution H to the equation (31) on the interval [T p , T ψ ], that satisfies the initial condition H(T p ) = 1e -λTp λ , and observe that H is smooth on the interval [T p , T ψ ]. Set F (t) = 1e λt H ′ (t) for each t ∈ [T p , T ψ ]

H 2 (

 2 T p ) := 1e -λTp λ and H ′ 2 (T p ) = e -λTp (1 -F 2 (T p )), the map H 2 is C 2 on the closed interval [T p , T ] and solves (22) on [T p , T ].

  1 )) exceeds s if and only if φ(t 1 ) > p * , that is, iff t 1 < T φ . It follows that γ 2 P (t 1 , t 1 ) ≤ γ 2 P (t 1 , t 2 ), and γ 2 P (t 1 , t 1 ) < γ 2 P (t 1 , t 2 ) if and only if t 1 < T φ . This observation allows us to pin down the best reply correspondence. Fix first a pure strategy t 1 of player 1. The map t 2 → γ 2 P (t 1 , t 2 ) coincides with π(•) on [0, t 1 ], and is constant on (t 1 , +∞], with γ 2 P (t 1 , t 1 +) ≥ γ 2 P (t 1 , t 1 ), and γ 2 P (t 1 , t 1 +) > γ 2 P (t 1 , t 1 ) if and only if t 1 < T φ . Since π(•) is single-peaked, with a maximum at T * * , it follows that the set B 2 (t 1 ) of pure best replies to t 1 is given by

  Proof. Set T 1 := max{t ∈ [T p , T ] : H ′′ (t) + λH ′ (t) ≤ 0}. From (22), one can check thatH ′′ (T p ) + λH ′ (T p ) < 0, hence T 1 > T p .We claim that T 1 = T . Assume to the contrary thatT 1 < T . One has H ′′ (t) ≤ -λH ′ (t) < 0 for each t i n[T p , T ], hence H ′ is decreasing on [T p , T 1 ],andH ′′ (T 1 ) < . Hence there exists T ' ∈ (T 1 , T ] such that H ′ is decreasing on [T 1 , T ′ ].is negative and decreasing on [T p , T ′ ]. On the other hand, H ′ is positive and decreasing, while H is positive and increasing on [T p , T ′ ]. Therefore,a(t)H ′ (t) + b(t)H(t) < a(T p )H ′ (T p ) + b(T p )H(T p ) (32)for each t ∈ (T p , T ′ ], and an easy computation shows that the right-hand side of (32) is equal to zero. Hence H ′′ + λH ′ is negative on [T 1 , T ′ ] -in contradiction with the definition of T 1 .Lemma 7 There exists t < T φ such that H ′ (t) = 0.Proof. Assume instead that H ′ (t) > 0 for each t ∈ [T p , T φ ). Integrating (22), there are c 1 , c 2 ∈ R, such thatH ′ (t) + λH(t) = c 1 + c 2

	t Tp	φ(x) -p * W (φ(x)) -s	H ′ (x)dx + λ
	F ′ 1 (t) H 1 (t) H We rewrite equation (22) as F ′ 2 (t) <
	α(t)(H	′′ (t) + λH ′ (t)) = a(t)H ′ (t) + b(t)H(t),

t) H i (t) = λφ(x)(p *ψ(x)) + (p *φ(x)) H ′ i (x) H i (x) , for t ∈ [T p , T ]. Hence, for fixed t, the ratio F ′ i (t) H i (t) is decreasing w.r.t. the ratio 1 -F i (t) H i (t) . Since Ω(t) > 0 for t ∈ [T p , T ), it follows that 2 (t) for t ∈ [T p , T ), and therefore, Ω(•) is increasing over the interval [T p , T ]. This contradicts the fact that Ω(T p ) > Ω( T ). This concludes the proof that there is at most one equilibrium. We now turn to existence. Since W (φ(x)) > s for x ∈ [T p , T φ ), there is a C 2 function H that solves (22) on the interval [T p , T φ ) and such that H(T p ) = 1e -λTp λ and H ′ (T p ) = e -λTp . Lemma 6 Let T ∈ [T p , T φ ) be such that H ′ (t) > 0 on [T p , T ]. Then H ′′ (t) + λH ′ (t) ≤ 0 for each t ∈ [T p , T ]. In particular, H ′ is decreasing over [T p , T ]. with a(t) := e λt (rsP(min(τ 1 , τ 2 ) ≥ t)λ(γs)P(R i = G, τ j ≥ t)), b(t) := λe λt (rsP(R i = G, τ j ≥ t)and α(t) > 0 for each t ∈ [T p , T φ ]. It can be shown that a(•) is positive and decreasing, while b(•) t Tp φ(x)(ψ(x)p * ) W (φ(x))s H(x)dx.

Thus, this is the case as soon as arms are negatively correlated and all information is public, or arms are positively correlated, and only actions are publicly observed.

In each of the four combinations but one, there is a unique symmetric equilibrium.

Formally, there exists no date at which one's belief jumps with probability one.

Throughout, and depending on the context, we use the word belief to denote either the posterior distribution of the pair of types, or the posterior probability assigned to one's arm being good. We hope that no confusion will arise.

Recall that τ j is not observed by player i.

Given the players's strategies, this is the positive probability event where no payoff has been received prior to t 1 .

Similar issues arise in the payoffs scenario as well.

Obviously, one cannot merge these two nodes into a single decision node, at which each player would be allowed to condition his time t-decision on the other player time t-decision. Indeed, circularity problems would arise, leading to the indeterminacy of the outcome.

Unless if risky arms are perfectly negatively correlated, in which case φ is constant. In what follows, we sometimes implicitly assume that arms are not perfectly negatively correlated, but all of our results cover that case as well.

That is, the smallest closed set in [0, +∞] that is assigned probability 1 by σ * .

That is, player j getting a payoff, or dropping out, depending on the scenario.

Recall that P is the benchmark decision problem in which the decision maker observes, on top of his own risky arm, whether the other risky arm produces any payoff.

This overall payoff depends on the exact time τ 1 . However, since it is multiplied by a probability which is itself of the order of dt, only the leading term has to be kept.

If ρ < 0, one has T * * = T φ , hence this situation cannot possibly arise.

Note that the condition is set on ξ i (t) in the first case, and on ξ i (t) in the second case.

Details of the computation are available upon request.

Otherwise, by Lemma 3, and since S 1 = S 2 , t would be a common atom of σ 1 and of σ 2 -contradicting Lemma 3.

Formal details are available from the authors.

strategies t 2 and T ψ are the same.

Appendix

We here provide all the formal details. We start with the analysis of the decision problem P

In Section A, and next complete the analysis of the payoffs scenario. The much longer section B contains the analysis of the actions scenario.

A The decision problem P

We adopt the convention that the decision maker in P is player 2. We recall that the expected payoff induced in P by the pure strategy t is denoted by π(t), and is given by π(t) = E e -rτ 1 W (ψ(τ 1 ))1 τ 1 ≤min(τ 2 ,t) +γE e -rτ 2 1 τ 2 <min(τ 1 ,t) +se -rt P(min(τ 1 , τ 2 ) ≥ t), for t ∈ R + .

Thus, for every two dates t < t ′ , one has π(t ′ )π(t) = E e -rτ 1 W (ψ(τ 1 ))1 t≤τ 1 <min(τ 2 ,t ′ ) + γE e -rτ 2 1 t≤τ 2 <min(τ 1 ,t ′ ) (6) +s e -rt ′ P(min(τ 1 , τ 2 ) ≥ t ′ )e -rt P(min(τ 1 , τ 2 ) ≥ t) .

We will prove that there is a unique optimal strategy, T * * , which is obtained as the unique solution to the equation λφ(T * * ) (W (ψ(T * * ) + γ -2s) = rs.

In addition, T * * ≥ T φ , and T * * > T φ if and only if ρ > 0.

The proof is organized in several steps.

Claim 8

There exists an optimal strategy.

Proof of the claim. By dominated convergence, π is continuous over R + , and has a limit when t → +∞, which we denote π(+∞). Letting t ′ → +∞ in (6), one gets

The first expectation in the right-hand side of ( 8) is at most γe -rt P(t ≤ τ 1 < +∞), while the second one is at most γe -rt P(t ≤ τ 2 < +∞). The probabilities P(t ≤ τ 1 < +∞) = P(t ≤ τ 2 < +∞) converge to zero as t → +∞, while P(min(τ 1 , τ 2 ) ≥ t) has a positive limit. Thus, π(+∞) < π(t) for every large t. It follows that π admits (at least) one maximum.

Proof. We prove that the first statement implies the second one. Fix x ∈ I. For y > x in I, the equality γ i A (x, σ j ) = γ i A (y, σ j ) writes 0 = 1 yx s e -r(y-x) P(θ j ≥ y, τ i ≥ y | min(θ j , τ i ) ≥ x) -1 +E W (φ(θ j ))e -r(θ j -x) 1 θ j <min(y,τ i ) | min(θ j , τ i ) ≥ x +γE e -r(τ i -x) 1 τ i <min(y,θ j ) | min(θ j , τ i ) ≥ x .

Taking the limit y ց x in the previous equality, since t → W (φ(t))e -r(t-x) is C 1 , and by Facts 1, 2, 3 and 3b, one obtains

Write N(x, y) := P(τ i ≥ x, θ j ∈ [x, y)), and D(x) := P(τ i ≥ x, θ j ≥ x). With these notations, equation ( 23) rewrites

We introduce the function Fj (t) := P(θ j ≤ t | R j = G), which is related to F j by the identity

With these notations at hand, one has N(x, y) = q GG e -λx + q GB ( Fj (y) -Fj (x)) + q GB e -λx + q BB (F j (y) -F j (x)).

Using the relation between Fj and F j , and the continuity of F j , equation (24) implies that F j is C 1 on the interval I, and therefore, Fj is C 1 as well, hence

On the other hand, D(x) is given by

B.4 Proof of Theorem 1 and of Proposition 2

We here specialize to the case ρ > 0. We start with the proof of Proposition 2.

B.4.1 Proof of Proposition 2

We let (σ 1 , σ 2 ) be an equilibrium, and let t 0 ∈ S i . Assume for a moment that ξ i (t 0 ) > p * . Since ρ > 0, this implies that ξ i (t+) > p * throughout some neighborhood of t 0 . Hence, by Fact 6, the map t → γ i A (t, σ j ) is increasing over some neighborhood, which is in contradiction with t 0 ∈ S i . Therefore, ξ i (t 0 ) ≤ p * . Since ρ > 0, this implies φ(t 0 ) ≤ p * . By Equation ( 14), this implies that t → γ i A (•, σ j ) is continuous at t 0 , so that γ i A (t 0 , σ j ) = γ i A (σ i , σ j ). In that case, γ i A (t 0 , σ j ) = γ i A (σ i , σ j ) and ξ i (t 0 ) = ξ i (t 0 +). We next claim that ξ i (t 0 ) = p * . Assume by way of contradiction that ξ i (t 0 ) < p * . We claim that γ i A (t 0 , σ j ) < γ i A (t, σ j ), provided t < t 0 is close enough to t 0 . This will yield the desired contradiction. Since φ(t 0 ) < p * , the quantity 1 e -rt P(θ

Since x → e -r(x-t) is C 1 , and by Facts 1 through 3, this expression has the same limit as

The latter expression is equal to the difference between the payoff in P induced by the policy t 0t, when starting with a belief of ξ i (t), and s, divided by t 0t. Since ξ i (t 0 ) < 0, the limit is therefore negative, as desired. Therefore, ξ i (t 0 ) = p * . We next prove that σ j ({t 0 }) = 0, so that ξ i (t 0 +) = ξ i (t) = p * , as desired. Assume to the contrary that σ j ({t 0 }) > 0. By Lemma 3, one has σ i ([t 0ε, t 0 ]) = 0 for some ε > 0. On the other hand, and since ξ i is discontinuous, one has ξ i (t 0 +) > p * . Using Fact 6, this implies that σ i ([t 0 , t 0 + ε]) = 0 for ε > 0 small enough. Hence σ i assigns probability zero to some neighborhood of t 0 . But this is in contradiction with t 0 ∈ S i .

B.4.2 Proof of Theorem 1

The proof is organized as follows. We first assume the existence of a symmetric equilibrium and show uniqueness. We next prove existence.

Let (σ, σ) be a symmetric equilibrium, with support S.

Lemma 4

The set S is equal to

viewing such a strategy σ as a probability distribution over S, we equivalently describe it by the conditional probability of dropping at t, σ({t})/σ([0, t)).

Given ω ∈ Ω, we define S 1 (ω) = {u 1 , . . . , u K } and S 2 (ω) = {v 1 , . . . , v K }.

Given ω ∈ Ω, we define inductively x 1 (ω), . . . , x K-1 (ω) as follows. We set x 1 (ω) = 0 if v 1 = u 1 (= T p ). If v 1 > u 1 , we define x 1 (ω) ∈ [0, 1] by the requirement that, if the strategy σ 1 of player has a support in S 1 (ω) and drops with probability x 1 (ω) at u 1 , then the belief ξ 2 (v 1 ) held by player 2 at v 1 is equal to p * . Similarly, x k (ω) is set to 0 if v k = u k , and otherwise defined by the condition that ξ 2 (v k ) = p * , if σ 1 has a support in S 1 (ω) and drops with probability x l (ω) at time u l (l ≤ k).

Somewhat symmetrically, the values y l (ω) (l = 1, . . . , K -1) are defined inductively in such a way that the belief ξ 1 (u k ) (k > 1) held by player 1 is equal to p * if σ 2 drops with probability y l (ω) at dates v l , l = 1, . . . , k -1.

Whether or not all dates in ω are distinct, one then has ξ 1 (u k ) = ξ 2 (v k ) = p * , for each k ∈ {1, . . . , K}. In addition, x k (ω) and y k (ω) lie in [0, 1] and are continuous in ω.

We introduce some additional piece of notation. Given ω ∈ Ω, and ũk

we denote by ω|ũ k the point in Ω obtained when substituting ũk to u k .

We next proceed by defining a continuous map from Ω into itself. It will be obtained as the composition of several maps α k (ω) (k > 1) and β k (ω) (k < K).

We start with α k (ω). Let ω ∈ Ω be given.

, and for any ũk ∈ (v k-1 , v k ), we denote by ∆ 2 k (ũ k ) the difference in player 2's expected payoffs induced by the strategies v k-1 and v k , when facing the strategy of player 1 which drops out with probability x l (ω|ũ k ) at time u 1 , . . . , u k-1 , ũk , . . .. We claim that ∆ 2 k (•) is decreasing over the open interval (v k-1 , v k ). To see this, let u k < ūk be given in (v k-1 , v k ). We denote by σ u k 1 and σ ūk 1 the corresponding two strategies of player 1. Using the strategy v k rather than v k-1 , when facing either σ u k 1 or σ ūk 1 , may yield a different outcome only if min(τ 2 , θ 1 ) ≥ v k-1 , and we henceforth assume that this event holds. The continuation payoff is then equal to s when using v k-1 . We will prove that the continuation payoff of strategy v k is higher when facing σ u k 1 than when facing σ ūk 1 . We proceed in several steps. Note first that to ensure a belief of p * at time v k , it must be that x k (ω|u k ) > x k (ω|ū k ). Next, everything else being kept constant, the expected payoff of player 2 (induced by the strategy v k ) is linear in the probability that player 1 drops out at u k (or at ūk ). In addition, the higher this probability, the more informative if player 1's decision at time u k , and therefore, the higher this payoff.

This implies that it will be sufficient to prove the following statement. (Conditional on τ 2 > v k-1 , θ 1 > v k-1 ), the expected payoff induced by the strategy v k is higher when facing a strategy that stops with probability one at u k (in the absence of payoffs) than when facing a strategy that stops with probability one at ūk . Note that the outcomes induced by these two profiles are identical, as soon as either τ 1 < u k or τ 2 < u k . Hence, we may and will condition on the event min(τ 1 , τ 2 ) ≥ u k . In that event, the continuation payoff induced by the first profile as of time u k is equal to s, since player 2 drops out at u k , immediately following player 1. On the other hand, under the second profile, player 2 will either drop out at ūk if τ 1 , τ 2 ≥ ūk , or remain active then, possibly forever. If ūk differs from u k by an infinitesimal amount of time, and using the same arguments as in earlier proofs, this expected payoff is approximately equal to

Using the fact that u k ≥ T p , so that p(t) ≤ p * , it is not difficult to show that this payoff does not exceed s, at least when ρ is sufficiently close to one, and the outside option s is close enough to zero.

This concludes the proof that ∆ 2 k is decreasing wrt ũk . Continuity of ∆ 2 k on the open interval (v k-1 , v k ) is easy to check. Finally, observe that, on the one hand, ∆ 2 k (•) > 0 when ũk is very close to v k-1 (the proof follows that of Fact 1), while ∆ 2 k (•) < 0 when ũk is very close to v k . The reason there is that in the limit ũk → v k , player 2 will drop out at time v k iff τ 2 ≥ v k . Hence there is a unique value, ûk , such that ∆ 2 k (û k ) = 0, and we set α k (ω) = (ω|û k ). The definition of β k (ω) slightly differs, to reflect the disymmetry between the two players. Let ω ∈ Ω be given. If u k = u k+1 , we set β k (ω) = u k . Otherwise, we set β k (ω) = (ω|v k ), where vk ∈ (u k , u k+1 ) is the unique point such that the expected payoffs induced by the two strategies u k and u k+1 against the strategy that drops out with probability y l (ω|v k ) at time v 1 , . . . , v k-1 , vk , . . ..

The map α

T ψ , and that equilibrium requirements are met by the strategies that drop out respectively with probability x l (ω * ) at time u * l , and with probability y l (ω * ) at time v * l .