Game Theory and Related Topics

Pescara - October 2-4 2003


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Talks Titles and Abstracts


Speaker: E Lehrer

Title: Information and its value in zero sum games
by Ehud Lehrer and Dinah Rosenberg

Abstract:
In a Bayesian game players play an unknown game. Before the game starts some players may receive some signal through an information structure regarding the specific game about to be played. In a one-sided repeated zero sum games we characterize when one information structure is better for the maximizer than another. We also characterize the functions that are value of information function in symmetric information games and in repeated one sided information games.


Speaker: P.Battigalli

Title: Rationalization and incomplete information
by Pierpaolo Battigalli e Marciano Siniscalchi

Abstract:
We analyse a family of solution procedures, akin to rationalizability, for models of strategic interaction with incomplete information. These procedures do not rely on the specification of an epistemic type space à la Harsanyi, but can take as given (and commonly believed) some exogenous restrictions on first order beliefs.

More precisely, we consider a multistage game structure with observable actions specifying, for each player, a set of possible payoff types, i.e. pieces of private information about preferences over consequences and about the relationship between actions (terminal histories) and consequences. The utility of each player is a function of the terminal history and of the vector of payoff types. Each players has beliefs about the types and strategies of his opponents, that are updated as the play unfolds. Beliefs at different histories are related to each other by Bayes rule whenever possible, hence they form a conditional probability system (CPS). Such CPS are the first order (conditional) beliefs of a player.

The solution procedure iteratively eliminates payoff type-strategy pairs for each player, taking as given a set of restrictions of players’ first order beliefs, represented by an array D of subsets of conditional probability systems (one subset for each payoff type of player). The solution, called D-rationalizability, embodies the following epistemic assumptions involving initial interactive beliefs as well as interactive belief revision (a formal proof can be found in the working paper version of Battigalli and Siniscalchi (2002)):

(A0) Each player is rational (i.e. maximizes his conditional expected utility at each relevant history, given his payoff type) and has first order beliefs satisfying the restrictions D.

(A1) Each player believes (A0) whenever possible.

(A2) Each player believes the conjunction (A0)&(A1) whenever possible.

(A,n+1) Each player believes the conjunction (A0)&(A1)&(A2)& &(An) whenever possible.

In static games, these assumptions simply amounts to common certainty of rationality and of the restrictions D. With no exogenous restrictions on beliefs, the procedure iteratively deletes pairs (q,s) such that s is dominated for q (iterated interim dominance). In dynamic games, the solution includes a form of forward induction reasoning (we focus on forward induction because it plays a prominent role in the analysis of many interesting economic models with incomplete information).

The game of incomplete information described above does not correspond to a Bayesian game in the sense of Harsanyi (1967-68). To obtain a Bayesian game we have to augment the model with an (implicit) description of the possible hierarchies of belief: beliefs about opponents’ payoff types, beliefs about such beliefs, and so on. An Harsanyi type encodes a player’s payoff type as well as his hierarchy of beliefs about other players’ types. Harsanyi (1967-68) points out that a Bayesian game is essentially isomorphic to a game with asymmetric information about an initial chance move selecting vectors of Harsanyi types (players may hold different prior beliefs about the chance move). A Bayesian equilibrium corresponds to a Nash equilibrium of the latter game. More explicitly, an equilibrium of a Bayesian game is an array of behavioural functions (one for each player) mapping Harsanyi types into strategic choices, such that each player holds correct conjectures about the choice that each possible Harsanyi type of each opponent would make, and maximizes his expected payoff given his type and his conjecture. We call Bayesian equilibrium model a combination of Bayesian game and equilibrium behavioural functions

We relate D-rationalizability to Bayesian equilibrium and other known equilibrium concepts. In particular, we prove the following results.

Proposition A. (i) In static games, D-rationalizability exactly characterises the set of outcomes (combinations of payoff types and strategies) that may occur in any Bayesian equilibrium model consistent with the restrictions D. (ii) In dynamic games, strong D-rationalizability refines this set of outcomes.

Next we focus on a particular kind of restrictions. To motivate them, consider the following situation. For each player-role there is a large population of individuals who are drawn and matched at random with individuals from other populations. Populations are heterogeneous with respect to payoff types and beliefs. Let z denote the statistical distribution of combinations of payoff types and terminal histories resulting from the many games played and suppose the statistic z becomes public. Players’ beliefs are stable if they are consistent with distribution z (in the sense that, for each player j, all the conditional frequencies of opponents’ types and actions implicit in z coincide with the corresponding conditional beliefs of player j). This yields a set of restrictions on first order beliefs and a corresponding z-rationalizability procedure. Similarly, it makes sense to consider the Bayesian equilibrium model consistent with z. Finally, we may wish to check whether z is a self-confirming equilibrium distribution on the terminal nodes of the resulting extensive form game where the prior on players’ payoff types is the marginal of z. [On self-confirming equilibria see Fudenberg and Levine, (1993). We consider self-confirming equilibria with unitary beliefs of the game where each payoff type corresponds to a distinct player.] We say that distribution z is feasible if it is generated by some strictly positive product prior on payoff types and a vector of behavioural strategies (vectors of locally randomised, type-dependent choices).

Proposition B. Suppose that distribution z is feasible. (i) z is a self-confirming equilibrium distribution if and only if there is a Bayesian equilibrium model consistent with z. (ii) If the set of z-rationalizable outcomes is nonempty, then z is a self-confirming equilibrium distribution.

Proposition B(i) holds because we are considering strong assumptions about observability. Suppose that players only observes (ex post) their realized payoffs. Then the analog of Proposition B(i) would not hold. Proposition B(ii) follows from Propositions A(ii) and B(i). It says that strong rationalizability yields a (forward induction) refinement of the self-confirming equilibrium concept.

Finally, we focus on signalling games and obtain a characterization of the Iterated Intuitive Criterion of Cho and Kreps (1987) (applied to self-confirming equilibrium distributions):

Proposition C. Fix a feasible distribution z on the terminal nodes of a signalling game (i.e. a distribution on Sender types, messages and responses). z is a self-confirming equilibrium satisfying the Iterated Intuitive Criterion if and only if the set of z-rationalizable outcomes is non-empty.

References

BATTIGALLI, P. and M. SINISCALCHI (2002): Strong Belief and Forward Induction Reasoning,Journal of Economic Theory, 106, 356-391.

CHO, I.-K. and D. KREPS (1987): Signaling Games and Stable Equilibria, Quarterly Journal of Economics, 102, 179-221.

FUDENBERG, D. and D.K. LEVINE (1993): Self-Confirming Equilibrium,Econometrica, 61, 523-545.

HARSANYI, J. (1967-68): Games of Incomplete Information Played by Bayesian Players. Parts I, II, III, Management Science, 14, 159-182, 320-334, 486-502


Speaker: M.Dall'Aglio

Title: Maximin share and envy in fair-division problems
(joint work with Theodore P. Hill)

Abstract:

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values of probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envy-minimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.


Speaker: P.Hernandez

Title: Costly communication in repeated interactions
(joint work with O.Grossner and A.Neyman)

Abstract:
We introduce a model of dynamic interactions with asymmetric information and costly communication. One player, the forecaster, has superior information to another player, the agent, concerning the realizations of a stream of states of nature.

A repeated game takes place between the sequence, the forecaster, and the agent. The agent chooses at each stage an action from a finite set that depends on the past history. The forecaster's stage decisions may depend not only on past history, but also on future realizations of nature. Hence, there are two aspects in the forecaster's stage decisions. First, this player can take strategic decisions that may affect both player's stage payoff. Second, the forecaster may choose messages from a set that will be subsequently observed by the agent. Our model encompasses these two aspects in a unified action set for the forecaster.

Our model allows player's stage payoffs to depend arbitrarily on their actions and on the state of nature. This includes the particular case in which the forecaster's action set is separable in a strategic decision space and a message space, and in which payoffs do not depend on the second.

In order to achieve cooperative outcomes in this repeated game, it may be necessary for the forecaster to use actions to inform the agent of future realized states of nature. Using information theory, we measure the amount of information sent by the forecaster to the agent at each stage, and the amount of information used by the agent in the course of the game.

Intuitively, during repetitions of the game, the total information used by the agent cannot exceed the total information received. This can be easily formalized and expressed using our measures for information. This in turn has implications on the set of empirical distributions over triples (state of nature, forecaster's actions, agent's actions) that are attainable by some strategies of the players in the repeated game. We express this constraint via a single formula, that we call the information constraint.

On the other hand, we prove that, given any distribution $Q$ over this triple that fulfills the information constraint, there exists a communication scheme between the forecaster and the agent such that the induced distribution is $Q$. Therefore, our results establish that the set of achievable empirical distributions given the communication constraints of the game is fully characterized by the information constraint.

Given any payoff function, the set of feasible payoffs in the repeated game can be computed by taking the image of the set of feasible distributions. Therefore, the information constraint also characterizes the set of feasible payoffs for all possible payoff functions.

We provide applications of the above approach to team problems and to situation with non-common objectives. When both the forecaster and the agent share common preferences, we characterize the best expected payoff that their team can guarantee. When preferences are not aligned, we characterize the set of equilibrium payoffs of the repeated game.


Speaker: J.Sakovics

Title: A Dynamic Theory of Holdup
(joint work with Yeon-Koo Che)

Abstract:

The holdup problem arises when parties negotiate to divide the surplus generated by their relationship specific investments.We study this problem in a dynamic model of bargaining and investment which, unlike the stylized static model, allows the parties to continue to invest until they agree on the terms of trade.The investment dynamic overturns the conventional wisdom dramatically. First , the holdup problem need not entail underinvestment when the parties are sufficiently patient. Second, inefficiencies can arise unambigously in some cases, but they are not caused by the sharing of surplus per se but rather by a failure of an individual rationality constraint.


Speaker: Marco Li Calzi

Title: The Pearson system of utility functions
(Marco Li Calzi and Annamaria Sorato)

Abstract:

This paper describes a technique to parameterize and estimate utility functions from elicited utilities. Inspired by the Pearson system of distributions, we suggest a four-parameter representation for the risk aversion function which embeds the HARA equation. The associated utility functions can have only four shapes: concave, convex, S-shaped, and reverse S-shaped. This makes the resulting family ideally suited for both expected utility and prospect theory. We also describe a simple method to estimate the parameters of the utility functions, which is faster and easier to implement than fitting by mean residual error minimization.


Speaker: N.Vieille

Title: About games of timing


Speaker: M.Marinacci

Title: "Stable Cores of Large Games"
with Luigi Montrucchio. August 2003.

Abstract: We consider two fundamental solution concepts for TU games, cores and von Neumann-Morgenstern stable sets. Stable sets are in general not easy to handle, as there are typically many of them and they are not easy to find. This has greatly limited the use of stable sets, despite theirconceptual appeal. In contrast, the core is a much simpler set to handle and it has gained a central importance in all applications. When the two solution concepts coincide, we have an ideal situation, in which the conceptual appeal of stable sets and the simplicity of cores are combined.

In this paper we consider continuous exact games and for them we provide a condition under which such an ideal situation occurs and the two fundamental solution concepts are equivalent. We then show that this condition is satisfied by important classes of games, such as convex games and exactmarket games. In this way we extend and unify several recent results for infinite games. [an error occurred while processing this directive]