Information and the Existence of Stationary Markovian Equilibrium
We describe conditions for the existence of a stationary Markovian equilibrium when total production or total endowment is a random variable. Apart from regularity assumptions, there are two crucial conditions: (i) low information—agents are ignorant of both total endowment and their own endowments when they make decisions in a given period, and (ii) proportional endowments—the endowment of each agent is in proportion, possibly random, to the total endowment. When these conditions hold, there is a stationary equilibrium. When they do not hold, such an equilibrium need not exist.
Construction of Stationary Markov Equilibria in a Strategic Market Game
This paper studies stationary noncooperative equilibria in an economy with fiat money, one nondurable commodity, countably many time periods, no credit or futures market, and a measure space of agents -- who may differ in their preferences and in the distributions of their (random) endowments. These agents are immortal, and hold fiat money as a means of hedging against the random fluctuations in their endowments of the commodity. In the aggregate, these fluctuations offset each other, and equilibrium prices are constant. We carry out an equilibrium analysis that focuses on distribution of wealth, on consumption, and on price formation. A careful analysis of the one-agent, infinite-horizon optimization problem, and of the invariant measure for the associated optimally controlled Markov chain, leads by aggregation to a stationary noncooperative or competitive equilibrium. This consists of a price for the commodity and of a distribution of wealth across agents which, under appropriate simple strategies for the agents, stay fixed from period to period and preserve the basic quantities of the model. We hope that, in future work, we shall be able to address additional features of the model treated here, such as borrowing and lending at appropriate (endogenously determined) interest rates, the endogenous production of the commodity, overlapping generations of agents, and bankruptcy and treatment of creditors.
Strategy and Manipulation in Medieval Elections
There are many goals in developing electoral protocols, including a desire for a system which is transparent, in that it is clear what the rule or procedure to follow is; nonmanipulable, in that it is not in a person’s best interest to misrepresent their preferences; honest, in the sense that it elects the ‘right’ candidate; and not open to strategizing, i.e., bribery or collusion. However, these desiderata are in tension with each other: Often, transparent electoral procedures are the least strategy resistant, and many honest procedures encourage manipulation. Thus a balance between these different goals must be sought. In modern times, since the seminal result on vote manipulation, the Gibbard-Satterthwaite Theorem [5, 6], much attention has been devoted to developing voting rules where manipulation is never in the best interest of the voters [4] or which are computationally too complex for the average bounded agent to be able to manipulate [1]. This focus on computational aspects of electoral methods is one of the hallmarks of modern studies on voting. But pursuit of these goals is not restricted to modern times: Those participating in elections in the Middle Ages also sought transparency, non-manipulability, honesty, and strategyproofness in so far as these properties can be consistently expressed in a single procedure. However, given the lack of computational sophistication in the Middle Ages, alternate approaches were needed in order to promote honesty, discourage strategizing, etc. These approaches can be classified as either external (constraints introduced outside of the electoral procedure, such as incentives for coming to consensus quickly) or internal (constraints introduced within the electoral procedure, such as voting rules which cannot be manipulated without adverse effects, or which are too difficult for the average bounded agent to manipulate). Surveying examples of both approaches in the context of medieval ecclesiastical and secular elections provides an interesting comparison to modern electoral procedures. Elections in the Middle Ages were used for the same reasons that they are today: To select suitable candidate(s) for a particular office, duty, or obligation. However, it is important to note that the term electio was used in the Middle Ages in a broader sense than our modern ‘election’. Its primary sense was ‘selection’ or ‘choice’, and only secondarily ‘election’ in the modern sense. Thus, many records which purportedly discuss elections are not discussing elections of the type which interests us. We can identify four categories of medieval electoral processes: (1) Election by an external authority having no direct interest in the election; (2) Indirect election, where electors name other electors who then select or elect the officials; (3) Election by lot; and (4) Election by ballot. Elections of the first and third types are generally computationally uninteresting; the first type corresponds to dictatorial voting rules, and the third type collapses to probability theory. In general, interesting voting methods are found only in the fourth type, election by ballot, though they can also occur in indirect election.