Online research in philosophy

We consider the following signaling game. Nature plays first from the set {1, 2}. Player 1 (the Sender) sees this and plays from the set {A, B}. Player 2 (the Receiver) sees only Player 1’s play and plays from the set {1, 2}. Both players win if Player 2’s play equals Nature’s play and lose otherwise. Players are told whether they have won or lost, and the game is repeated. An urn scheme for learning coordination in this game is as follows. Each node of the desicion tree for Players 1 and 2 contains an urn with balls of two colors for the two possible decisions. Players make decisions by drawing from the appropriate urns. After a win, each ball that was drawn is reinforced by adding another of the same color to the urn. A number of equilibria are possible for this game other than the optimal ones. However, we show that the urn scheme achieves asymptotically optimal coordination.

Issues that arise in using game theory to model national security problems are discussed, including positing nation-states as players, assuming that their decision makers act rationally and possess complete information, and modeling certain conflicts as two-person games. A generic two-person game called the Conflict Game, which captures strategic features of such variable-sum games as Chicken and Prisoners'' Dilemma, is then analyzed. Unlike these classical games, however, the Conflict Game is a two-stage game in which each player can threaten to retaliate — and carry out this threat in the second stage — if its opponent chose noncooperation in the first stage.Conditions for the existence of different pure-strategy Nash equilibria, or stable outcomes, are found, and these results are extended to situations in which the players can select mixed strategies (i.e., make probabilistic threats or choices). Although the Conflict Game sheds light on the rational foundations underlying arms races, nuclear deterrence, and other strategic situations, more detailed assumptions are required to tie this generic game to specific conflicts.

In certain finite extensive games with perfect information, Cristina Bicchieri (1989) derives a logical contradiction from the assumptions that players are rational and that they have common knowledge of the theory of the game. She argues that this may account for play outside the Nash equilibrium. She also claims that no inconsistency arises if the players have the minimal beliefs necessary to perform backward induction. We here show that another contradiction can be derived even with minimal beliefs, so there is no paradox of common knowledge specifically. These inconsistencies do not make play outside Nash equilibrium plausible, but rather indicate that the epistemic specification must incorporate a system for belief revision. Whether rationality is common knowledge is not the issue.

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like rationality at a node and knowledge at a node. We distinguish between the game theorist's and the players' own theory of the game. The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory.

Payoff dominance, a criterion for choosing between equilibrium points in games, is intuitively compelling, especially in matching games and other games of common interests, but it has not been justified from standard game-theoretic rationality assumptions. A psychological explanation of it is offered in terms of a form of reasoning that we call the Stackelberg heuristic in which players assume that their strategic thinking will be anticipated by their co-player(s). Two-person games are called Stackelberg-soluble if the players' strategies that maximize against their co-players' best replies intersect in a Nash equilibrium. Proofs are given that every game of common interests is Stackelberg-soluble, that a Stackelberg solution is always a payoff-dominant outcome, and that in every game with multiple Nash equilibria a Stackelberg solution is a payoff-dominant equilibrium point. It is argued that the Stackelberg heuristic may be justified by evidentialist reasoning.

We will study game trees as representations of rational choice and as representations of player preferences, and promises as public announcements of genuine intentions. Promises in a game change what players know about the preferences of other players. They can be modelled as operations that change a given game into a different game where players know more about the effects of their strategies.

The traditional solution concept for noncooperative game theory is the Nash equilibrium, which contains an implicit assumption that players’ probability distributions satisfy t probabilistic independence. However, in games with more than two players, relaxing this assumption results in a more general equilibrium concept based on joint beliefs (Vanderschraaf, 1995). This article explores the implications of this joint-beliefs equilibrium concept for two kinds of conflictual coordination games: crisis bargaining and public goods provision. We find that, using updating consistent with Bayes’ rule, players’ beliefs converge to equilibria in joint beliefs which do not satisfy probabilistic independence. In addition, joint beliefs greatly expand the set of mixed equilibria. On the face of it, allowing for joint beliefs might be expected to increase the prospects for coordination. However, we show that if players use joint beliefs, which may be more likely as the number of players increases, then the prospects for coordination in these games declines vis-à -vis independent beliefs.

Human behavior can be analyzed using game theory models. Complex games may involve different rules for different players and may allow players to change identity (and therefore, rules) according to complex contingencies. From this perspective, mating behaviors can be viewed as strategic “plays” in a complex “mating game,” with players varying tactics in response to changes in the game's payoff matrix.

In his last dialogue, the Laws, Plato views citizens in the polis as players in a game. Just as contemporary game theory, Plato considers games to be states of strategic interaction. Yet the game of the Laws differs from those of game theory in one important respect. Where game theory assumes that players are rational--that they choose strategies, or rules for taking action at each instant of a game, in order to maximize payoffs--Plato explores the conditions under which rationality, as game theory defines it, is possible. Plato thus agrees with game theory that rational, maximizing behavior is a necessary constituent of civic order, but not that it may simply be assumed. He concurs that rationality can describe the behavior of citizens, but not under any circumstances. It is the task of politics in Plato's city to prepare the conditions for rationality. Only once politics has done its work is maximizing, utilitarian behavior possible. Yet political preparation of the game of the city is exactly what contemporary game theory assumes away. The Athenian, who leads the discussion, describes the political preparation for rationality as "this moderate old man's game concerning laws." Political preparation of the game of the city is itself a game, because it is never possible to escape strategic interaction. It is, however, a second-order game whose play paves the way for the first-order game of lawful strategic interaction. The second-order, political game at once completes the analysis of rationality and lays the groundwork for rationality to operate. The politics of the Laws is a game, but one whose actions and players differ from the actions and players of the first-order game. The action of the political game is lawgiving. The players are the gods and god-like men who serve as lawgivers. The second-order, political game also has its own distinct rationality, linked to a payoff that is qualitatively different from the utilitarian payoffs of the first-order game. The Athenian calls the rationality of the second-order game "intelligence." He calls the payoff associated with it "joy," as distinct from the "benefits" associated with utilitarian rationality. The players in the second-order game seek to maximize joy, not benefit. Joy is the experience of play. The payoff of players in the second-order game is the game itself, not a benefit collateral to playing it. The second-order game is a "true" game, one that the players enter in order to play, not to get utilitarian payoffs.

It is usually assumed in game theory that agents who interact strategically with each other are rational, know the strategies open to other agents as well as their payoffs and, moreover, have common knowledge of all the above. In some games, that much information is sufficient for the players to identify a "solution" and play it. The most commonly adopted solution concept is that of Nash equilibrium. A Nash equilibrium is defined a combination of strategies, one for each player, such that no player can profit from a deviation from his strategy if the opponents stick to their strategies. Nash equilibrium is taken to have predictive power, in the sense that in order to predict how rational agents will in fact behave, it is enough to identify the equilibrium patterns of actions. Barring the case in which players have dominant strategies, to play her part in a Nash equilibrium a player must believe that the other players play their part, too. But an intelligent player must immediately realize that she has no ground for this belief. Take the case of a one-shot, simultaneous game. Here all undominated strategies are possible choices, and the beliefs supporting them are possible beliefs, even if this game has a..