in a one - shot prisoner 's dilemma ( pd ) game , the two players have to choose between c and d ( to cooperate resp . to defect ) . following the notation in rapoport and chammah ( 1965 ) , the payoff matrix is given by(1 ) in which the four payoff variables represent the reward for mutual cooperation r , the sucker 's payoff s , the temptation to defect t , and the punishment for mutual defection p. payoffs satisfy the inequalities t > r > p > s , such that defection is a dominant strategy , but mutual cooperation is preferred over mutual defection . in addition to these inequalities , we shall also assume that(2)2r > t+s , such that mutual cooperation is unanimously preferred from a group perspective.1 in such cases , experimental evidence suggests that many players want to achieve conditional cooperation . they are willing to play c , provided the co - player also plays c ( see , e.g. , fehr and fischbacher , 2003 ; yamagishi et al . , 2005 ) . thus , players either have to trust their co - player , or else use their dominant strategy . the situation is different for an iterated pd game ( ipd ) . diverse  folk theorems state that any feasible and individually rational outcome can be sustained as an equilibrium if the probability  of a further round is sufficiently large . such outcomes can be enforced in various ways , and under a wide range of circumstances ( see , e.g. , friedman , 1971 ; aumann , 1981 ; aumann and shapley , 1994 ; fudenberg and maskin , 1986 ; kalai , 1990 ; myerson , 1991 ; mailath and olszewski , 2011 ) . if subjects can make binding commitments ahead of the game , then analogous results can be obtained even for one - shot games ( kalai et al . , 2010 ) . experimental research has uncovered considerable heterogeneity in human social preferences ( colman , 1995 ; kagel and roth , 1997 ; camerer , 2003 ) , and a similar variety can be found among the strategies that are played in the ipd ( milinski and wedekind , 1998 ; dal b and frchette , 2011 ; fudenberg et al . , 2012 ) . players who in the one - shot game would opt for conditional cooperation should be willing to engage in  partner strategies . such strategies aim for an average payoff r per round , which necessarily provides the same payoff r for the co - player ; should the co - player not go along , however , then the co - player 's payoff will be less than r. thus , a partner strategy appeals to the co - player 's self - interest in order to further the own self - interest . it is fair , and provides an incentive for the co - player to also be fair . in contrast , some players tend to view their co - player as a rival , rather than a partner . the main purpose , for such competitive players , is to do better , or at least as well as the other player . a preference for dominating the co - player is particularly likely in the context of a game , which often has antagonistic connotations . the aim of the present manuscript is two - fold : first , we are going to characterize all memory - one strategies which are either competitive , or partner strategies in the sense above . we emphasize that players using such strategies can enforce their preferences against all comers , since we impose no restrictions on the strategies used by the co - players . for partner strategies , the corresponding results in the limiting case of an ipd without discounting have been obtained within the last two years ( akin , 2013 ; stewart and plotkin , 2013 , 2014 ) . here , we are going to extend the theory by allowing for discount factors 1 ( which may either be interpreted as the constant continuation probability of having another round , or as the players ' common discount rate on future payoff streams ) . the recent progress was stimulated by the unexpected discovery of so - called zero - determinant ( zd ) strategies , a class of memory - one strategies enforcing a linear relationship between the payoffs of the two players , irrespective of the co - player 's strategy ( press and dyson , 2012 ) . in particular , zd strategies can fix the co - player 's payoff to an arbitrary value between p and r ; or ensure that the own surplus ( over the maximin value p ) is twice as large as the co - player 's surplus ; etc . also for zd strategies , we are going to extend the theory to the case when future payoffs are discounted , and 1 . our article does not focus on equilibrium behavior ( in particular , we do not aim to explore which payoffs rational players can achieve ) . instead , we define some interesting properties that a player 's strategy may have ( e.g. , being competitive ) ; and then we are going to characterize all memory - one strategies that have the respective property ( independent of whether such a strategy can be sustained as an equilibrium ) . thereby , we do not make any assumptions on the behaviors of the co - players ( e.g. , we do not require them to play best responses , or to follow a predefined equilibrium path ) . nevertheless , there are natural connections between several of the described strategy classes and equilibrium behavior , and in that case we will discuss these connections in detail . in the discussion , we will briefly review the previous development , and in particular the relevant findings in evolutionary game theory ( stewart and plotkin , 2012 , 2013 , 2014 ; adami and hintze , 2013 ; hilbe et al . , 2013a , 2013b ; akin , 2013 ; szolnoki and perc , 2014a , 2014b ; wu and rong , 2014 ) . in a nutshell , these findings say that in the context of populations of adapting players , partner strategies do well , whereas competitive strategies fare poorly . we consider the standard setup of an ipd with perfect monitoring . in each round , that is , they choose an action from their respective action set ai={c , d } , with i{i , ii}. the outcome of a given round t can then be described by an action profile ata = aiaii . after each round , both players observe the chosen action profile , and they receive the respective payoffs as specified in the payoff matrix ( 1 ) . the round t history is a vector ht=(a0,a1,  ,at1)at , and the set of possible histories is the union h=t=0at , with the initial history a0 being defined as the null set a0={}. a strategy for player i is a rule that tells the player how to act after any possible history ; that is , a strategy is a map i : h(ai ) , where (ai ) denotes the set of probability distributions over the action set ai.2 for given strategies of the two players , let va(t ) denote the probability that the resulting action profile played in round t is a{cc , cd , dc , dd}. for convenience , we use the following vector notation:(3)v(t)=(vcc(t),vcd(t),vdc(t),vdd(t))gi=(r , s , t , p)gii=(r , t , s , p ) using this notation , we can write the players ' expected payoffs in round t as i(t)=giv(t ) and ii(t)=giiv(t ) . for a discount factor <1 , the expected payoffs of the repeated game can then be defined by the abelian means(4)i=(1)t=0ti(t)=giv , and similarly ii = giiv , where v=(vcc , vcd , vdc , vdd ) refers to the ( abelian ) mean distribution(5)v=(1)t=0tv(t ) . in the limiting case =1 , the payoff per round is given by the cesaro mean(6)i = lim1+1t=0i(t ) ( if this limit exists ) , and a similar expression for ii.3 a theorem by frobenius states that if the cesaro mean exists , it is the limit of the abelian mean , for 1 . in the following , we will sometimes focus on players who only take the decisions in the previous round into account . definition 1a strategy  is a memory - one strategy if (ht)=(ht ) for all histories ht=(a0, a strategy  is a memory - one strategy if (ht)=(ht ) for all histories ht=(a0, ,at1 ) and ht=(a0,  ,at1 ) with t , t1 and at1=at1 . for rounds t1 , the move of a memory - one player is therefore solely determined by the action profile played in the previous round ( in particular , we note that such players do not condition their behavior on the round number , as sometimes considered in models of bounded recall , e.g. mailath and olszewski , 2011 ) . such memory - one strategies can be written as a 5-tuple p=(pcc , pcd , pdc , pdd;p0 ) . the continuation vector p:=(pcc , pcd , pdc , pdd ) denotes the conditional probabilities to cooperate in rounds t1 , depending on the outcome of the previous round ( slightly abusing notation , we let the first letter in the subscript refer to the player 's own action in the previous round , and the second letter to the co - player 's action . using this convention , we ensure that the interpretation of a memory - one strategy does not depend on whether the player acts as player i or as player ii , see nowak and sigmund , 1995 ) . examples of memory - one strategies include alld=(0,0,0,0;0 ) , tit for tat ( 1,0,1,0;1 ) , or win - stay , lose - shift ( 1,0,0,1;1 ) , see sigmund ( 2010 ) for a comprehensive discussion . when both players apply a memory - one strategy , the resulting mean distribution v can be calculated explicitly ( nowak and sigmund , 1995 ) : if player i uses the memory - one strategy p=(pcc , pcd , pdc , pdd;p0 ) against a player ii with memory - one strategy q=(qcc , qcd , qdc , qdd;q0 ) , then(7)v=(1)v(0)(i4m)1 , where v(0)=(p0q0 ,  p0(1q0 ) ,  ( 1p0)q0 ,  ( 1p0)(1q0 ) ) is the initial distribution , i4 is the 44 identity matrix , and m is the transition matrix of the markov chain,(8)m=(pccqccpcc(1qcc)(1pcc)qcc(1pcc)(1qcc)pcdqdcpcd(1qdc)(1pcd)qdc(1pcd)(1qdc)pdcqcdpdc(1qcd)(1pdc)qcd(1pdc)(1qcd)pddqddpdd(1qdd)(1pdd)qdd(1pdd)(1qdd ) ) . but even if only one of the players is using a memory - one strategy p , there is still a powerful relationship between p and the resulting mean distribution v. lemma 1suppose player i applies a memory - one strategy 
 p , and let the strategy of player ii be arbitrary , but fixed.(i)in the case with discounting 
 ( <1 ) , let 
 v 
 denote the mean distribution of the repeated game . then(9)(pcc1)vcc+(pcd1)vcd+pdcvdc+pddvdd=(1)p0 , 
 or in vector notation , 
 ( pg0)v=(1)p0 , where 
 g0=(1,1,0,0).(ii)in the case without discounting , we have(10)lim1+1t=0(pg0)v(t)=0 . 
 in particular , if the cesaro mean distribution 
 v 
 exists , 
 ( pg0)v=0 . suppose player i applies a memory - one strategy 
 p , and let the strategy of player ii be arbitrary , but fixed.(i)in the case with discounting 
 ( <1 ) , let 
 v 
 denote the mean distribution of the repeated game . then(9)(pcc1)vcc+(pcd1)vcd+pdcvdc+pddvdd=(1)p0 , 
 or in vector notation , 
 ( pg0)v=(1)p0 , where 
 g0=(1,1,0,0).(ii)in the case without discounting , we have(10)lim1+1t=0(pg0)v(t)=0 . in particular , if the cesaro mean distribution 
 v 
 exists , 
 ( pg0)v=0 . in the case with discounting 
 ( <1 ) , let 
 v 
 denote the mean distribution of the repeated game . then(9)(pcc1)vcc+(pcd1)vcd+pdcvdc+pddvdd=(1)p0 , 
 or in vector notation , 
 ( pg0)v=(1)p0 , where 
 g0=(1,1,0,0 ) . in the case without discounting , we have(10)lim1+1t=0(pg0)v(t)=0 . 
 in particular , if the cesaro mean distribution 
 v 
 exists , 
 ( pg0)v=0 proofsuppose <1 , and let qi(t ) denote the probability that player i cooperates in round t. then qi(t)=g0v(t ) and qi(t+1)=pv(t ) . it follows that w(t):=qi(t+1)qi(t ) is given by(11)w(t)=(pg0)v(t ) . multiplying each w(t ) by ( 1)t and summing up over t=0, ( 11),(13)(1)t=0tw(t)=(1)t=0t(pg0)v(t)(pg0)v as both limits need to coincide , we have confirmed eq . ( 9 ) . for the case without discounting , an analogous calculation as in eq . ( 12 ) yields(14)1+1t=0w(t)0 , whereas eq . it follows that the limit of 1+1t=0(pg0).v(t ) for  exists and equals zero . suppose <1 , and let qi(t ) denote the probability that player i cooperates in round t. then qi(t)=g0v(t ) and qi(t+1)=pv(t ) . it follows that w(t):=qi(t+1)qi(t ) is given by(11)w(t)=(pg0)v(t ) . multiplying each w(t ) by ( 1)t and summing up over t=0, ( 11),(13)(1)t=0tw(t)=(1)t=0t(pg0)v(t)(pg0)v as both limits need to coincide , we have confirmed eq . ( 9 ) . for the case without discounting , an analogous calculation as in eq . ( 12 ) yields(14)1+1t=0w(t)0 , whereas eq . ( 13 ) becomes(15)1+1t=0w(t)=1+1t=0(pg0)v(t ) . it follows that the limit of 1+1t=0(pg0).v(t ) for  exists and equals zero . it is worthwhile to stress the generality of lemma 1 : it neither makes any assumption on the strategy used by the co - player , nor does it depend on the specific payoff constraints of a prisoner 's dilemma . in the limiting case =1 , lemma 1 allows a geometric interpretation : the mean distribution v ( if it exists ) is orthogonal to pg0 ( see akin , 2013 ) . definition 2a player 's strategy is nice , if the player is never the first to defect . a player 's strategy is cautious if the player is never the first to cooperate . a player 's strategy is nice , if the player is never the first to defect . a player 's strategy is cautious if the player is never the first to cooperate . for memory - one strategies , nice strategies , the strategy tft 
 ( 1,0,1,0;1 ) is nice , whereas the defector 's strategy alld 
 ( 0,0,0,0;0 ) is cautious . lemma 2if 
 2r > t+s , then payoffs satisfy 
 i+ii2r 
 if and only if 
 i=ii = r 
 ( which for 
 <1 
 holds if and only if both players are nice ) . similarly , if 
 2p < t+s , then 
 i+ii2p 
 if and only if 
 i=ii = p 
 ( which for 
 <1 
 is equivalent to both players being cautious ) . if 
 2r > t+s , then payoffs satisfy 
 i+ii2r 
 if and only if 
 i=ii = r 
 ( which for 
 <1 
 holds if and only if both players are nice ) . similarly , if 
 2p < t+s , then 
 i+ii2p 
 if and only if 
 i=ii = p 
 ( which for 
 <1 
 is equivalent to both players being cautious ) . this requires both players to cooperate in every round ( if =1 , it only requires the players to cooperate in almost every round ) . similarly , for a prisoner 's dilemma with 2p < t+s , the inequality i+ii2p implies vdd=1 . this requires both players to cooperate in every round ( if =1 , it only requires the players to cooperate in almost every round ) . similarly , for a prisoner 's dilemma with 2p < t+s , the inequality i+ii2p implies vdd=1 . definition 3(i)a partner strategy for player i is a nice strategy such that , irrespective of the co - player 's strategy,(16)i < rii < r.(ii)a competitive strategy for player i is a strategy such that , irrespective of the co - player 's strategy,(17)iii . ( i)a partner strategy for player i is a nice strategy such that , irrespective of the co - player 's strategy,(16)i < rii < r.(ii)a competitive strategy for player i is a strategy such that , irrespective of the co - player 's strategy,(17)iii . a partner strategy for player i is a nice strategy such that , irrespective of the co - player 's strategy,(16)i < rii < r . a competitive strategy for player i is a strategy such that , irrespective of the co - player 's strategy,(17)iii . the definition of partner strategies implies that these strategies are best replies to themselves , and thus they are nash equilibria . even more , because condition ( 16 ) is equivalent to ( iir)(ir ) , we can conclude due to lemma 2 that ( iir)(i=ii = r ) . thus , no matter which best reply the co - player applies , a player with a partner strategy will always obtain the mutual cooperation payoff r. on the other hand , players with a competitive strategy always obtain at least the co - player 's payoff . it is easy to see that for <1 a competitive strategy needs to be cautious ( otherwise the focal player would be outcompeted by an alld - player ) . in the limiting case =1 , competitiveness is closely related to the concept of being unbeatable , as introduced by duersch et al . a strategy for player i is unbeatable , if against any co - player and for any number of rounds , the payoff differential t=0(ii(t)i(t ) ) is bounded from above ( in particular , if the average payoffs per round converge to i and ii , then iii ) . proposition 1for a player i with a nice memory - one strategy 
 p , the following are equivalent:(i)p 
 is a partner strategy;(ii)if the co - player uses either alld or the strategy 
 ( 0,1,1,1;0 ) , then 
 ii < r;(iii)the two inequalities 
 b1<0 
 and 
 b2<0 
 hold , with(18)b1=(tr)pdd(rp)(1pcd)+(1)(tr)b2=(tr)pdc(rs)(1pcd)+(1)(tr ) . for a player i with a nice memory - one strategy 
 p , the following are equivalent:(i)p 
 is a partner strategy;(ii)if the co - player uses either alld or the strategy 
 ( 0,1,1,1;0 ) , then 
 ii < r;(iii)the two inequalities 
 b1<0 
 and 
 b2<0 
 hold , with(18)b1=(tr)pdd(rp)(1pcd)+(1)(tr)b2=(tr)pdc(rs)(1pcd)+(1)(tr ) . p 
 is a partner strategy ; if the co - player uses either alld or the strategy 
 ( 0,1,1,1;0 ) , then 
 ii < r ; the two inequalities 
 b1<0 
 and 
 b2<0 
 hold , with(18)b1=(tr)pdd(rp)(1pcd)+(1)(tr)b2=(tr)pdc(rs)(1pcd)+(1)(tr ) . then the definition of partner strategies implies that i=ii = r . since all players use memory - one strategies , this would require that everyone cooperates after mutual cooperation , which is neither true for alld=(0,0,0,0;0 ) nor for the strategy ( 0,1,1,1;0).(ii )  ( iii)against a player using a nice memory - one strategy p ( with p0=pcc=1 ) , the payoff of an alld co - player is given by(19)ii=(1)t+pppcd+tpdd1+(pddpcd ) . we note that this payoff is also defined when =1 , because pcd<1 ( otherwise p would satisfy p0=pcc = pcd=1 , and player i would always cooperate . in that case , an alld co - player would receive t > r , which is ruled out by ( ii ) ) . elementary algebra yields(20)b1=(1+(pddpcd))(iir ) , in particular , b1 has the same sign as iir . on the other hand , if the co - player uses the strategy ( 0,1,1,1;0 ) , the co - player 's payoff is(21)ii=(1)t+s+((1)rs)pcd+(t+r)pdc1+2(pdcpcd)+pdc , and(22)b2=(1+2(pdcpcd)+pdc)(iir ) . therefore , b2 has the same sign as iir.(iii )  ( i)suppose that b1<0 and b2<0 , and that iir . we need to show that ii=i = r . as iir=(giir1)v , with 1=(1,1,1,1 ) , we note that iir is equivalent to(23)(tr)vcd(rs)vdc(rp)vdd0 . using the linear equations 1v=1 and ( pg0)v=(1 ) ( lemma 1 with p0=pcc=1 , since the memory - one strategy is nice ) , we calculate vcd as a function of vdc and vdd:(24)vcd=(1(1pdc)))vdc+(1(1pdd))vdd(1pcd). the denominator of vcd is positive , as b1<0 implies pcd<1 . plugging ( 24 ) into ( 23 ) and multiplying both sides with ( 1pcd) shows that iir if and only if(25)b2vdc+b1vdd0 , with b1 and b2 as defined in ( 18 ) . thus , the assumptions b1<0 and b2<0 indicate that vdc = vdd=0 , and by ( 24 ) that vcd=0 . we conclude that vcc=1 , and therefore i=ii = r , i.e. then the definition of partner strategies implies that i=ii = r . since all players use memory - one strategies , this would require that everyone cooperates after mutual cooperation , which is neither true for alld=(0,0,0,0;0 ) nor for the strategy ( 0,1,1,1;0).(ii )  ( iii)against a player using a nice memory - one strategy p ( with p0=pcc=1 ) , the payoff of an alld co - player is given by(19)ii=(1)t+pppcd+tpdd1+(pddpcd ) . we note that this payoff is also defined when =1 , because pcd<1 ( otherwise p would satisfy p0=pcc = pcd=1 , and player i would always cooperate . in that case , an alld co - player would receive t > r , which is ruled out by ( ii ) ) . elementary algebra yields(20)b1=(1+(pddpcd))(iir ) , in particular , b1 has the same sign as iir . on the other hand , if the co - player uses the strategy ( 0,1,1,1;0 ) , the co - player 's payoff is(21)ii=(1)t+s+((1)rs)pcd+(t+r)pdc1+2(pdcpcd)+pdc , and(22)b2=(1+2(pdcpcd)+pdc)(iir ) . therefore , b2 has the same sign as iir.(iii )  ( i)suppose that b1<0 and b2<0 , and that iir . , we note that iir is equivalent to(23)(tr)vcd(rs)vdc(rp)vdd0 . using the linear equations 1v=1 and ( pg0)v=(1 ) ( lemma 1 with p0=pcc=1 , since the memory - one strategy is nice ) , we calculate vcd as a function of vdc and vdd:(24)vcd=(1(1pdc)))vdc+(1(1pdd))vdd(1pcd). the denominator of vcd is positive , as b1<0 implies pcd<1 . plugging ( 24 ) into ( 23 ) and multiplying both sides with ( 1pcd) shows that iir if and only if(25)b2vdc+b1vdd0 , with b1 and b2 as defined in ( 18 ) . thus , the assumptions b1<0 and b2<0 indicate that vdc = vdd=0 , and by ( 24 ) that vcd=0 . we conclude that vcc=1 , and therefore i=ii = r , i.e. then the definition of partner strategies implies that i=ii = r . since all players use memory - one strategies , this would require that everyone cooperates after mutual cooperation , which is neither true for alld=(0,0,0,0;0 ) nor for the strategy ( 0,1,1,1;0 ) . against a player using a nice memory - one strategy p ( with p0=pcc=1 ) , the payoff of an alld co - player is given by(19)ii=(1)t+pppcd+tpdd1+(pddpcd ) . we note that this payoff is also defined when =1 , because pcd<1 ( otherwise p would satisfy p0=pcc = pcd=1 , and player i would always cooperate . in that case , an alld co - player would receive t > r , which is ruled out by ( ii ) ) . elementary algebra yields(20)b1=(1+(pddpcd))(iir ) , in particular , b1 has the same sign as iir . on the other hand , if the co - player uses the strategy ( 0,1,1,1;0 ) , the co - player 's payoff is(21)ii=(1)t+s+((1)rs)pcd+(t+r)pdc1+2(pdcpcd)+pdc , and(22)b2=(1+2(pdcpcd)+pdc)(iir ) . as iir=(giir1)v , with 1=(1,1,1,1 ) , we note that iir is equivalent to(23)(tr)vcd(rs)vdc(rp)vdd0 . using the linear equations 1v=1 and ( pg0)v=(1 ) ( lemma 1 with p0=pcc=1 , since the memory - one strategy is nice ) , we calculate vcd as a function of vdc and vdd:(24)vcd=(1(1pdc)))vdc+(1(1pdd))vdd(1pcd). the denominator of vcd is positive , as b1<0 implies pcd<1 . plugging ( 24 ) into ( 23 ) and multiplying both sides with ( 1pcd) shows that iir if and only if(25)b2vdc+b1vdd0 , with b1 and b2 as defined in ( 18 ) . thus , the assumptions b1<0 and b2<0 indicate that vdc = vdd=0 , and by ( 24 ) that vcd=0 . we conclude that vcc=1 , and therefore i=ii = r , i.e. for example , tft is a partner strategy if and only if >trtp and >trrs , whereas wsls is a partner strategy if and only if >trrp and >trts , which is a sharper condition . in analogy to the definition of partner strategies , one may define a mild partner strategy for player i as a nice strategy such that ir implies iir , irrespective of the co - player 's strategy.4 for memory - one strategies , the characterization of mild partner strategies is analogous to the characterization of partner strategies ( only the strict inequalities in proposition 1 need to be replaced by weak inequalities ) . the existence of an equilibrium with individually rational payoffs ( i,ii ) in the ipd is typically shown by applying trigger strategies  any deviation from the equilibrium path is punished with relentless defection ( as for example in friedman , 1971).5 the following corollary states that trigger strategies are , in some sense , the most effective means to enforce a cooperative equilibrium in the ipd . corollary 1for a given prisoner 's dilemma and a given continuation probability  , there exists a memory - one partner strategy if and only if the trigger strategy grim 
 = ( 1,0,0,0;1 ) 
 is a partner strategy . proofthe two quantities b1 and b2 in proposition 1 are minimal for pcd = pdc = pdd=0 . thus , if there is a memory - one strategy that meets the inequalities b1<0 and b2<0 , then the corresponding inequalities are also met by grim . for a given prisoner 's dilemma and a given continuation probability  , there exists a memory - one partner strategy if and only if the trigger strategy grim 
 = ( 1,0,0,0;1 ) 
 is a partner strategy . the two quantities b1 and b2 in proposition 1 are minimal for pcd = pdc = pdd=0 . thus , if there is a memory - one strategy that meets the inequalities b1<0 and b2<0 , then the corresponding inequalities are also met by grim . from corollary 1 , we may also conclude that partner strategies exist if and only if >trtp ( this condition for the existence of fully cooperative equilibria has been previously derived by roth and murnighan , 1978 ; stahl , 1991 ) . let us next give a characterization of competitive memory - one strategies : proposition 2suppose player i applies the memory - one strategy 
 p. then the following are equivalent:(i)p 
 is competitive.(ii)if the co - player uses either alld or the strategy 
 ( 0,0,0,1;0 ) , then 
 iii.(iii)the entries of 
 p 
 satisfy 
 p0=pp=0 
 and 
 (pcd+pdc)1 . suppose player i applies the memory - one strategy 
 p. then the following are equivalent:(i)p 
 is competitive.(ii)if the co - player uses either alld or the strategy 
 ( 0,0,0,1;0 ) , then 
 iii.(iii)the entries of 
 p 
 satisfy 
 p0=pp=0 
 and 
 (pcd+pdc)1 . if the co - player uses either alld or the strategy 
 ( 0,0,0,1;0 ) , then 
 iii . proof(i )  ( ii)follows immediately from the definition , a competitive strategy yields iii against any co - player.(ii )  ( iii)if player ii applies alld , then an explicit calculation of payoffs yields(26)iii=(ts)((1)p0+pdd)1+(pddpcd ) , and thus iii implies p0=pdd=0 . similarly , if player ii applies the strategy ( 0,0,0,1;0 ) , we obtain ( using p0=pdd=0):(27)iii=(ts)(1(pcd+pdc))1+(1(1+)pcd+pdc ) . this is non - negative if and only if (pcd+pdc)1.(iii )  ( i)by lemma 1 and as pdd = p0=0,(28)pdcvdc=(1pcc)vcc+(1pcd)vcd . using the inequality pdc1pcd ( i )  ( ii)follows immediately from the definition , a competitive strategy yields iii against any co - player.(ii )  ( iii)if player ii applies alld , then an explicit calculation of payoffs yields(26)iii=(ts)((1)p0+pdd)1+(pddpcd ) , and thus iii implies p0=pdd=0 . similarly , if player ii applies the strategy ( 0,0,0,1;0 ) , we obtain ( using p0=pdd=0):(27)iii=(ts)(1(pcd+pdc))1+(1(1+)pcd+pdc ) . this is non - negative if and only if (pcd+pdc)1.(iii )  ( i)by lemma 1 and as pdd = p0=0,(28)pdcvdc=(1pcc)vcc+(1pcd)vcd . using the inequality pdc1pcd follows immediately from the definition , a competitive strategy yields iii against any co - player . if player ii applies alld , then an explicit calculation of payoffs yields(26)iii=(ts)((1)p0+pdd)1+(pddpcd ) , and thus iii implies p0=pdd=0 . similarly , if player ii applies the strategy ( 0,0,0,1;0 ) , we obtain ( using p0=pdd=0):(27)iii=(ts)(1(pcd+pdc))1+(1(1+)pcd+pdc ) . this is non - negative if and only if (pcd+pdc)1 . by lemma 1 and as pdd pdc1pcd , this leads to(29)(1pcd)vdc(1pcc)vcc+(1pcd)vcd , or equivalently ( 1pcd)(vdcvcd)(1pcc)vcc . this implies vdcvcd . as a consequence , iii=(gigii)v=(ts)(vdcvcd)0 . 2 shows the space of partner strategies ( and the space of competitive strategies ) as subsets of the nice memory - one strategies ( cautious memory - one strategies ) , respectively . one can also define the dual properties , and derive the corresponding characterizations : a strategy for player i is said to be submissive if payoffs always satisfy iii , irrespective of the strategy of player ii ; and a cautious strategy for player i is said to be requiting if i > p implies ii > p ( see fig . 1 for a schematic representation of these strategy classes ) . the corresponding characterizations are : proposition 3suppose player i applies the memory - one strategy 
 p. then the following are equivalent:(i)p 
 is submissive;(ii)if the co - player uses either allc or the strategy 
 ( 0,1,1,1;1 ) , then 
 iii;(iii)the entries of 
 p 
 satisfy 
 p0=pcc=1 
 and 
 (1pcd)+(1pdc)1 . suppose player i applies the memory - one strategy 
 p. then the following are equivalent:(i)p 
 is submissive;(ii)if the co - player uses either allc or the strategy 
 ( 0,1,1,1;1 ) , then 
 iii;(iii)the entries of 
 p 
 satisfy 
 p0=pcc=1 
 and 
 (1pcd)+(1pdc)1 . if the co - player uses either allc or the strategy 
 ( 0,1,1,1;1 ) , then 
 iii ; the entries of 
 p 
 satisfy 
 p0=pcc=1 
 and 
 (1pcd)+(1pdc)1 . then , for a player i with a cautious memory - one strategy 
 p , the following are equivalent:(i)p 
 is requiting;(ii)if the co - player uses either allc or the strategy 
 ( 0,0,0,1;1 ) , then 
 ii > p;(iii)the two inequalities 
 b1>0 
 and 
 b2>0 
 hold , with(30)b1=(rp)pdc+(ps)pcc(ps),b2=(tp)pdc+(ps)pcd(ps ) . then , for a player i with a cautious memory - one strategy 
 p , the following are equivalent:(i)p 
 is requiting;(ii)if the co - player uses either allc or the strategy 
 ( 0,0,0,1;1 ) , then 
 ii > p;(iii)the two inequalities 
 b1>0 
 and 
 b2>0 
 hold , with(30)b1=(rp)pdc+(ps)pcc(ps),b2=(tp)pdc+(ps)pcd(ps ) . if the co - player uses either allc or the strategy 
 ( 0,0,0,1;1 ) , then 
 ii > p ; the two inequalities 
 b1>0 
 and 
 b2>0 
 hold , with(30)b1=(rp)pdc+(ps)pcc(ps),b2=(tp)pdc+(ps)pcd(ps ) . the previous results have highlighted how lemma 1 can be used to characterize several interesting strategy classes within the space of memory - one strategies ( for example , the strategies that allow a player to outcompete the opponent , or the strategies that provide incentives to reach the social optimum ) . in the following , we present another application of lemma 1 : there are strategies with which a player can unilaterally enforce a linear relationship between the players ' payoffs . definition 4a memory - one strategy p is said to be a zd strategy if there exist constants ,, such that(31)p=gi+gii+((1)p0)1+g0 . a memory - one strategy p is said to be a zd strategy if there exist constants ,, such that(31)p=gi+gii+((1)p0)1+g0 . proposition 5let 
 <1 , and suppose player i applies a memory - one strategy 
 p 
 satisfying eq . the same relation holds for 
 =1 , provided that the payoffs 
 i 
 and 
 ii 
 exist.6 let 
 <1 , and suppose player i applies a memory - one strategy 
 p 
 satisfying eq . the same relation holds for 
 =1 , provided that the payoffs 
 i 
 and 
 ii 
 exist.6 proofthis follows directly from lemma 1 , using the identities i = giv , ii = giiv , and 1=1v . this follows directly from lemma 1 , using the identities i = giv , ii = giiv , and 1=1v . we proceed with a slightly different representation of zd strategies , using the parameter transformation = , = , and =(1).7 under this transformation , zd strategies take the form(33)p=[(1)(1gi)+(gigii)](1)p01+g0 , and the enforced payoff relationship according to ( 32 ) becomes(34)ii=(i ) . ( 34 ) implies that the payoffs lie on a line segment intersecting the diagonal at some value  ( the payoff for the zd - strategy against itself ) and having a slope  ( see fig . 3 ) . players can not use zd strategies to enforce arbitrary payoff relationships of the form ( 34 ) : since the entries of the continuation vector p correspond to conditional probabilities ( and hence need to be in the unit interval ) , the parameters  ,  and  need to obey certain restrictions . definition 5for a given  , we call a payoff relationship ( ,)r2 
 enforceable if there are r and p0 such that each entry of the continuation vector p according to eq . we refer to the set of all enforceable payoff relationships as e. for a given  , we call a payoff relationship ( ,)r2 
 enforceable if there are r and p0 such that each entry of the continuation vector p according to eq . we refer to the set of all enforceable payoff relationships as e. proposition 6(i)the set of enforceable payoff relationships is monotonically increasing in the discount factor : if 
  , then 
 ee.(ii)there is a 
 <1 
 such that 
 ( ,)e 
 if and only if 
 1<<1 
 and(35)max{p , st1}min{r , ts1 } , 
 with at least one inequality in 
 ( 35 ) 
 being strict . ( i)the set of enforceable payoff relationships is monotonically increasing in the discount factor : if 
  , then 
 ee.(ii)there is a 
 <1 
 such that 
 ( ,)e 
 if and only if 
 1<<1 
 and(35)max{p , st1}min{r , ts1 } , 
 with at least one inequality in 
 ( 35 ) 
 being strict . the set of enforceable payoff relationships is monotonically increasing in the discount factor : if 
  , then 
 ee. there is a 
 <1 
 such that 
 ( ,)e 
 if and only if 
 1<<1 
 and(35)max{p , st1}min{r , ts1 } , 
 with at least one inequality in 
 ( 35 ) 
 being strict . proof(i)according to the definition , ( ,)e if and only if one can find r and p0 , such that the corresponding continuation vector p according to eq . ( 33 ) satisfies 0p1 , or equivalently,(36a)(1)(1p0)(1)(r)1(1)p0(36b)(1)(1p0)[(1)(s)+ts]1(1)p0(36c)(1)p0[(1)(t)+ts]+(1)p0(36d)(1)p0(1)(p)+(1)p0 we note that in ( 36a)(36d ) , the left hand side is monotonically decreasing in  , whereas the right hand side is monotonically increasing in . in particular , if the conditions ( 36 ) are satisfied for some 1 they are also satisfied for any .(ii)()suppose ( ,)e , and therefore the conditions ( 36 ) hold for appropriate parameters  and p0 . summing up the first inequality in ( 36a ) and the first inequality in ( 36d ) shows(37)1(1)(rp ) . similarly , by taking the inequalities in ( 36b ) and ( 36c ) , we get(38)1(1+)(ts ) . in particular , 0<(1 ) and 0<(1+ ) , and therefore >0 and 1<<1 . if none of the inequalities in ( 35 ) was strict , then ( 36a ) or ( 36b ) would require p0=1 , whereas ( 36c ) or ( 36d ) would require p0=0.()conversely , let 1<<1 , and suppose max{p , st1}<min{r , ts1}. then the inequalities ( 39 ) hold for any choice of >0 , with the first two inequalities being strict . in particular , we can choose a  sufficiently small such that each term on the right hand 's side of ( 39 ) is bounded from above by 1/2 . by setting p0=0 and choosing a  sufficiently close to one , it thus follows that all inequalities in ( 36 ) can be satisfied . an analogous argument holds when =min{r , ts1 } , in which case one needs to set p0=1 . ( i)according to the definition , ( ,)e if and only if one can find r and p0 , such that the corresponding continuation vector p according to eq . ( 33 ) satisfies 0p1 , or equivalently,(36a)(1)(1p0)(1)(r)1(1)p0(36b)(1)(1p0)[(1)(s)+ts]1(1)p0(36c)(1)p0[(1)(t)+ts]+(1)p0(36d)(1)p0(1)(p)+(1)p0 we note that in ( 36a)(36d ) , the left hand side is monotonically decreasing in  , whereas the right hand side is monotonically increasing in . in particular , if the conditions ( 36 ) are satisfied for some 1 they are also satisfied for any .(ii)()suppose ( ,)e , and therefore the conditions ( 36 ) hold for appropriate parameters  and p0 . summing up the first inequality in ( 36a ) and the first inequality in ( 36d ) shows(37)1(1)(rp ) . similarly , by taking the inequalities in ( 36b ) and ( 36c ) , we get(38)1(1+)(ts ) . in particular , 0<(1 ) and 0<(1+ ) , and therefore >0 and 1<<1 . moreover , the conditions ( 36 ) imply(39)0(1)(r)0[(1)(s)+ts]0[(1)(t)+ts]0(1)(p ) . since >0 and <1 , these conditions are equivalent to condition ( 35 ) . if none of the inequalities in ( 35 ) was strict , then ( 36a ) or ( 36b ) would require p0=1 , whereas ( 36c ) or ( 36d ) would require p0=0.()conversely , let 1<<1 , and suppose max{p , st1}<min{r , ts1}. then the inequalities ( 39 ) hold for any choice of >0 , with the first two inequalities being strict . in particular , we can choose a  sufficiently small such that each term on the right hand 's side of ( 39 ) is bounded from above by 1/2 . by setting p0=0 and choosing a  sufficiently close to one , it thus follows that all inequalities in ( 36 ) can be satisfied . an analogous argument holds when =min{r , ts1 } , in which case one needs to set p0=1 . according to the definition , ( ,)e if and only if one can find r and p0 , such that the corresponding continuation vector p according to eq . ( 33 ) satisfies 0p1 , or equivalently,(36a)(1)(1p0)(1)(r)1(1)p0(36b)(1)(1p0)[(1)(s)+ts]1(1)p0(36c)(1)p0[(1)(t)+ts]+(1)p0(36d)(1)p0(1)(p)+(1)p0 we note that in ( 36a)(36d ) , the left hand side is monotonically decreasing in  , whereas the right hand side is monotonically increasing in . in particular , if the conditions ( 36 ) are satisfied for some 1 they are also satisfied for any . ( )suppose ( ,)e , and therefore the conditions ( 36 ) hold for appropriate parameters  and p0 . summing up the first inequality in ( 36a ) and the first inequality in ( 36d ) shows(37)1(1)(rp ) . similarly , by taking the inequalities in ( 36b ) and ( 36c ) , we get(38)1(1+)(ts ) . in particular , 0<(1 ) and 0<(1+ ) , and therefore >0 and 1<<1 . moreover , the conditions ( 36 ) imply(39)0(1)(r)0[(1)(s)+ts]0[(1)(t)+ts]0(1)(p ) . since >0 and <1 , these conditions are equivalent to condition ( 35 ) . if none of the inequalities in ( 35 ) was strict , then ( 36a ) or ( 36b ) would require p0=1 , whereas ( 36c ) or ( 36d ) would require p0=0.()conversely , let 1<<1 , and suppose max{p , st1}<min{r , ts1}. then the inequalities ( 39 ) hold for any choice of >0 , with the first two inequalities being strict . in particular , we can choose a  sufficiently small such that each term on the right hand 's side of ( 39 ) is bounded from above by 1/2 . by setting p0=0 and choosing a  sufficiently close to one , it thus follows that all inequalities in ( 36 ) can be satisfied . an analogous argument holds when =min{r , ts1 } , in which case one needs to set p0=1 . suppose ( ,)e , and therefore the conditions ( 36 ) hold for appropriate parameters  and p0 . summing up the first inequality in ( 36a ) and the first inequality in ( 36d ) shows(37)1(1)(rp ) . similarly , by taking the inequalities in ( 36b ) and ( 36c ) , we get(38)1(1+)(ts ) . in particular , 0<(1 ) and 0<(1+ ) , and therefore >0 and 1<<1 . moreover , the conditions ( 36 ) imply(39)0(1)(r)0[(1)(s)+ts]0[(1)(t)+ts]0(1)(p ) . since >0 and <1 , these conditions are equivalent to condition ( 35 ) . if none of the inequalities in ( 35 ) was strict , then ( 36a ) or ( 36b ) would require p0=1 , whereas ( 36c ) or ( 36d ) would require p0=0 . conversely , let 1<<1 , and suppose max{p , st1}<min{r , ts1}. then the inequalities ( 39 ) hold for any choice of >0 , with the first two inequalities being strict . in particular , we can choose a  sufficiently small such that each term on the right hand 's side of ( 39 ) is bounded from above by 1/2 . by setting p0=0 and choosing a  sufficiently close to one , it thus follows that all inequalities in ( 36 ) can be satisfied . an analogous argument holds when =min{r , ts1 } , in which case one needs to set p0=1 . the first part of proposition 6 shows that a given linear payoff relationship of the form ( 34 ) is easier to enforce when players are sufficiently patient . as 1 , the limiting set of enforceable payoff relationships ( , ) there are various remarkable subclasses of zd strategies ( as depicted in fig . 3 and fig . , we encounter so - called equalizer strategies ( see boerlijst et al . , 1997 ; press and dyson , 2012 ) . by eq . ( 34 ) , player i can make use of such strategies to prescribe  as payoff for player ii . a player can thus determine the opponent 's payoff ( however , player i can not fix the own payoff , since this would require  to be unbounded , which is ruled out by proposition 6 ) . press and dyson ( 2012 ) also highlighted the class of extortion strategies ( with =p and 0<<1 ) . extortion strategies guarantee that the own  surplus over the minimax payoff p exceeds the opponent 's surplus by a factor of 1 . moreover , since >0 , the payoffs of the two players are positively related . hence , to maximize the own payoff , player ii needs to maximize player i 's payoff : the best response against an extortioner is to cooperate unconditionally . as a counterpart to extortioners , stewart and plotkin ( 2012 ) defined the class of generous strategies , which satisfy eq . ( 34 ) with =r and 0<<1 . players using a generous strategy shoulder a larger burden of the loss ( with respect to the social optimum r ) than their co - player . since >0 , they also ensure that the payoffs of the two players are aligned , thereby motivating the co - player to cooperate . finally , for games without discounting it was noted that strategies with =1 enforce i=ii ( for =1 , tft is an example of such a fair strategy , see press and dyson , 2012 ; hilbe et al . , 2014b ) . however , as proposition 6 shows , fair strategies cease to exist when future payoffs are discounted , and only approximately fair strategies ( with  close to one ) may be feasible . zd strategies can also be connected to the strategy classes discussed in the previous section . generous strategies , for example , are exactly the zd strategies which are submissive partner strategies ( in particular it follows that every generous strategy is a nash equilibrium of the ipd ) . on the other hand , for stage games with 2p < t+s ( which ensures i+ii2p ) , extortion strategies are precisely those zd strategies which are requiting and competitive . we note that herein , we have entirely focused on the repeated prisoner 's dilemma , due to the central role that this simple game situation takes in the literature on the evolution of cooperation ( rapoport and chammah , 1965 ; trivers , 1971 ; sugden , 1986 ; axelrod , 1984 ; sigmund , 2010 ) . however , the proofs of lemma 1 and proposition 5 did not require any assumptions on the payoff values ( and in the proof of proposition 6 , we have only made use of the assumptions r > p and t > s ) . moreover , for =1 , it was recently shown that similar results can also be obtained for stage games with 2 actions but n2 players ( hilbe et al . , 2014b ) . thus , while we believe that our results are most intuitive in the context of a prisoner 's dilemma , the mathematics can be extended to more general strategic situations . the recent development began with the paper of press and dyson ( 2012 ) introducing zd strategies for repeated games without discounting . in this context , press and dyson derived the linear relation ( 32 ) . their proof was based on a neat formula for the payoffs achieved if both players use memory - one strategies . press and dyson highlighted those zd strategies that fix the co - player 's payoff to a given value between p and r , as well as the sinister properties of extortion strategies . they also stressed the fact that more complex strategies ( based on larger memories , for instance ) are not able to profit from their sophistication to gain the upper hand . the intriguing aspects of zd strategies raised considerable attention ( see , e.g. , ball , 2012 ) . in the news section of the american mathematical society , it was stated that ' the world of game theory is currently on fire . ' a more skeptical view could be found among economists . the well - known folk theorem for repeated games states that trigger strategies can induce a rational co - player to agree to any feasible payoff pair above the minimax level p , by threatening to switch to relentless defection otherwise ( see aumann , 1981 ; kalai , 1990 ; fudenberg and maskin , 1986 , 1990 ) . seen from this angle , the progress consisted merely in displaying memory - one strategies with a similar power to enforce specific payoff pairs . however , there is a subtle difference : whereas the folk theorems are based on the assumption that players wish to maximize their payoffs , the results presented herein are independent of such an assumption . interpreted in this way , we have explored how much control player i can exert on the resulting payoffs without being sure about the motives of player ii . memory - one strategies able to fix the co - player 's payoff had already been derived in boerlijst et al . ( 1997 ) and sigmund ( 2010 ) , based on an approach different from that of press and dyson ( 2012 ) . ( 2013a ) to provide another derivation of ( 32 ) , not involving any determinants . it was substantially extended by akin ( 2013 ) to yield a general equation for the mean distribution of memory - one strategies when =1 . in this case , the mean distribution is understood in the sense of cesaro , and need not always exist . in lemma 1 , we have extended this approach to cover the case <1 , from which akin 's result for =1 immediately follows . the vector p consists of the conditional probabilities to play c in the next round (  is the probability that there is a next round ) , whereas g0 can be viewed as ' conditional probability ' to play c in the current round . in the limit of no discounting , lemma 1 states that no matter which strategy player ii is using , the limiting distribution v ( if it exists ) is on a hyperplane orthogonal to the difference of these two conditional probabilities . it was also akin ( 2013 ) who extended the investigations beyond the case of zd - strategies , to characterize partner strategies for =1 ( calling them ' good ' strategies , a term we feel is too general ) . in a comments article , stewart and plotkin ( 2012 ) introduced an example of a generous strategy , and showed that in a round robin tournament conducted after the fashion of axelrod ( 1984 ) , this generous strategy emerged as winner . stewart and plotkin also asked whether zd strategies were relevant for evolutionary game theory . in this context , one considers a population of players , each equipped with a strategy . the players are then allowed to imitate other strategies , preferentially those with a higher payoff ( see , e.g. , weibull , 1995 ; samuelson , 1997 ; hofbauer and sigmund , 1998 ; nowak , 2006 ; sandholm , 2010 ) . it is obvious that extortion strategies can not spread too much in such an evolutionary context ; if they become too common , they are likely to encounter their own , which bides ill . if player i obtains twice the surplus of ii , and ii twice the surplus of i , each surplus is zero . however , hilbe et al . ( 2013a ) showed that extortion strategies can pave the way for the emergence of cooperative strategies , similar to tft ( which , for =1 , can be regarded as a limiting case of an extortion strategy , press and dyson , 2012 ) . this catalytic role of extortion strategies has also been confirmed for games on networks , in which players only interact within a small neighborhood ( szolnoki and perc , 2014a , 2014b ; wu and rong , 2014 ) . overall , these studies confirm that extortionate strategies have problems to succeed within a population . however , if the games are played between members of two distinct populations  for instance , between hosts organisms and their symbionts  then extortion strategies can emerge in whichever population is slower to adapt ( hilbe et al . . the slower rate of evolution acts as a commitment device . in effect , the slowly evolving organism becomes the stackelberg leader in a sequential game , in which the slow player learns to adopt extortion strategies , whereas the faster evolving player learns to play the best response , and to cooperate unconditionally ( bergstrom and lachmann , 2003 ; damore and gore , 2011 ; gokhale and traulsen , 2012 ) . but even in a one - population setup , certain zd strategies prove successful : stewart and plotkin showed that evolutionary trajectories often visit the vicinity of generous strategies . the dynamics leads  from extortion to generosity ( the title of stewart and plotkin , 2013 ) . this is also confirmed , by analytical means based on adaptive dynamics , by hilbe et al . remarkably , stewart and plotkin ( 2013 , 2014 ) derived a characterization of all memory - one strategies which are robust in an evolutionary sense , for given population size n. this means that the replacement probability , as a resident strategy , is at most 1/n ( which is the probability to be replaced if the mutant is neutral ) . in the limit of weak selection , which roughly means that the choice between two strategies is only marginally influenced by payoff ( see nowak et al . , 2004 ) , all robust zd strategies need to be generous ( stewart and plotkin , 2013 ) . these predictions have also been tested in a recent behavior experiment , in which human subjects played against various computer opponents ( hilbe et al . , 2014a ) . although extortionate programs outcompeted their human co - players in every game , generous programs received , on average , higher payoffs against the human subjects than extortionate programs . humans were hesitant to give in to extortion ; although unconditional cooperation would have been their best response in all treatments , they only became more cooperative over time if their co - player was generous . intriguingly , if a player uses a generous strategy and the co - player does not go along , then the focal player will always shoulder a larger part of the loss ( with respect to the mutual cooperation payoff r ) . despite their forbearance , generous strategies do very well  which is not the least of the surprises offered by the iterated prisoner 's dilemma game .