Bounded rationality and correlated equilibria

We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability p such that all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality \(\overline{p}\) that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.     

Sharing beliefs about actions

We consider a strategic situation in which each player may not know the probability distribution governing the information structures of his opponents, and consequently his beliefs about opponents' action choices are represented by a set of probability measures. Suppose that beliefs of all the players are common knowledge. Then for any subset of players, the marginal beliefs of those players (about the action choices of their common opponents) must share at least one probability measure.     

Games of timing related to distribution of resources

The paper considers a class of zero-sum, two-person games which are related to distribution of resources. Each of the players is in possession of an amount of resource, to be distributed by him in the time interval [0, 1] according to an arbitrary measure. The payoff function is defined in such a manner that the games are a generalization of the so-called silent, nondiscrete duels. It is proven that these games have a value, and the optimal strategies for the players are found. The results of the paper bring to light new, essential elements, common to almost all games of timing on [0, 1].     

Inducing cooperation by reciprocative strategy in non-zero-sum games

In this paper, we investigate the use of reciprocative strategy to induce cooperative behavior in non-zero-sum games. Reciprocative behavior is defined mathematically in the context of a two-person non-zero-sum game in which both the players have a common set of pure strategies. Conditions under which mutual cooperative behavior results when one of the players responds optimally to reciprocative behavior by the other player are described. Also, the desirability of playing the reciprocative strategy is investigated by stating conditions under which reciprocative strategy by one of the players or by both the players leading to mutual cooperative behavior is a Nash equilibrium outcome.     

Small-parameter method for constructing approximate strategies in a class of differential games : PMM vol. 39, n 6, 1975, pp. 1006–1016

We examine a class of problems in which the pay-off is some function of the terminal state of a conflict-controlled system. When the opportunities of one of the players are small in relation with the opportunities of the other, we propose methods for constructing approximate optimal strategies of the players, based on solving the Bellman equation containing a small parameter. We have shown that the players' approximate optimal strategies can be constructed if the solutions of the corresponding optimal control problems are known. The error bounds for the methods are proved and examples are considered. The arguments used rely on the results in [1–6] on the theory of differential games and on [7–11] devoted to optimal control synthesis methods for systems subject to random perturbations of small intensity.     

Tractable approximate knowledge fusion using the Horn fragment of serial propositional dynamic logic

In this paper we investigate a technique for fusing approximate knowledge obtained from distributed, heterogeneous information sources. This issue is substantial, e.g., in modeling multiagent systems, where a group of loosely coupled heterogeneous agents cooperate in achieving a common goal. Information exchange, leading ultimately to knowledge fusion, is a natural and vital ingredient of this process. We use a generalization of rough sets and relations [30], which depends on allowing arbitrary similarity relations.The starting point of this research is [6], where a framework for knowledge fusion in multiagent systems is introduced. Agents’ individual perceptual capabilities are represented by similarity relations, further aggregated to express joint capabilities of teams. This aggregation, expressing a shift from individual to social level of agents’ activity, has been formalized by means of dynamic logic. The approach of Doherty et al. (2007) [6] uses the full propositional dynamic logic, which does not guarantee tractability of reasoning. Our idea is to adapt the techniques of Nguyen [26], [27] and [28] to provide an engine for tractable approximate database querying restricted to a Horn fragment of serial dynamic logic. We also show that the obtained formalism is quite powerful in applications.     

The principal-agent problem with adaptive players

In this study we show predictions made by the standard principal-agent theory may not hold when knowlege assumptions are relaxed. Conventional principal-agent models assume players are completely rational: they know their own and other player's utilities and probabilities of all states of nature. In reality, players must make decisions without such knowledge.We define a simple version of the principal-agent game and examine it using object-oriented computer simulation. Player learning is modeled with a statistical learning model. Our results show that even this simple game combined with standard learning assumptions results in complex behavior. Expectations of both the principal and the agents are crucial in determining the system outcomes. Expectations and lack of prior knowledge make it possible for the principal to converge on suboptimal behavior or not converge on a consistent behavior at all. The same attributes in the agents make it possible for the principal to drive expectations down and thus get higher effort for lower reward.This study contributes a more robust understanding of the principal-agent model and its application to incentive design.     

On some games of timing of mixed type

The paper considers a zero-sum, two-person game of timing on [0, 1] in which Players 1 and 2 behave as in the so-called discrete-fire duel and non-discrete fire duel, respectively. Player 1 is in possession of one action. Accuracy functions of the players are continuous and nondecreasing, from [0, 1] onto [0, 1]. The game is analysed in three versions and the form of optimal strategies for the players is found.     

On the non-emptiness of the one-core and the bargaining set of committee games

We study the committee decision making process using game theory. Shenoy  [15] introduced two solution concepts: the one-core and the bargaining set, and showed that the one-core of a simple committee game is nonempty if there are at most four players. We extend this result by proving that whether the committee is simple or not, as far as there are less than five players, the one-core is nonempty. This result also holds for the bargaining set.     

An alternative characterization of the weighted Banzhaf value

We provide a new characterization of the weighted Banzhaf value derived from some postulates in a recent paper by Radzik, Nowak and Driessen [7]. Our approach owes much to the work by Lehrer [4] on the classical Banzhaf value based on the idea of amalgamation of pairs of players and an induction construction of the value. Compared with the approach in [7] we consider two new postulates: a weighted version of Lehrer’s “2-efficiency axiom” [4] and a generalized “null player out” property studied in terms of symmetric games by Derks and Haller [2]. Received: December 1997/final version: October 1999     

Application of Atanassov’s I-fuzzy set theory to matrix games with fuzzy goals and fuzzy payoffs

We aim to extend some results in [6, 7, 8, 2] on two person zero sum matrix games (TPZSMG) with fuzzy goals and fuzzy payoffs to I-fuzzy scenario. Because the payoffs of the matrix game are fuzzy numbers, the aspiration levels of the players are fuzzy as well. It is reasonable to believe that there is some indeterminacy in estimating the aspiration levels of both players from their respective expected pay offs. This situation is modeled in the game using Atanassov??s I-fuzzy set theory. A new solution concept is proposed for such games and a procedure is outlined to obtain the degrees of suitability of the aspiration levels for each of the two players.     

Observing,reporting, and deciding in networks of sentences

In prior work [7] we considered networks of agents who have knowledge bases in first order logic, and report facts to their neighbors that are in their common languages and are provable from their knowledge bases, in order to help a decider verify a single sentence. In report complete networks, the signatures of the agents and the links between agents are rich enough to verify any decider?s sentence that can be proved from the combined knowledge base. This paper introduces a more general setting where new observations may be added to knowledge bases and the decider must choose a sentence from a set of alternatives. We consider the question of when it is possible to prepare in advance a finite plan to generate reports within the network. We obtain conditions under which such a plan exists and is guaranteed to produce the right choice under any new observations.     

Product rule wins a competitive game

We consider a game that can be viewed as a random graph process. The game has two players and begins with the empty graph on vertex set . During each turn a pair of random edges is generated and one of the players chooses one of these edges to be an edge in the graph. Thus the players guide the evolution of the graph as the game is played. One player controls the even rounds with the goal of creating a so-called giant component as quickly as possible. The other player controls the odd rounds and has the goal of keeping the giant from forming for as long as possible. We show that the product rule is an asymptotically optimal strategy for both players.

     

Expectations and fairness in a simple bargaining experiment

We evaluate two competing hypotheses that try to account for robust violations of the predictions of game theory in Ultimatum bargaining experiments. One popular hypothesis is that the subjects are motivated by considerations of fairness that are not modelled by traditional theory. The alternative hypothesis is that the subjects do not have common knowledge of the rationality, beliefs and motives of other players. Each hypothesis can explain existing data. We design several experiments to discriminate between these two hypotheses. The results provide strong support for the alternative hypothesis.     

Learning to be prepared

Behavioral economics provides several motivations for the common observation that agents appear somewhat unwilling to deviate from recent choices. More recent choices can be more salient than other choices, or more readily available in the agent’s mind. Alternatively, agents may have formed habits, or use rules of thumb. This paper provides discrete-time adjustment processes for strategic games in which players display such a bias towards recent choices. In addition, players choose best replies to beliefs supported by observed play in the recent past. We characterize the limit behavior of these processes by showing that they eventually settle down in minimal prep sets (Voorneveld in Games Econ Behav 48:403–414, 2004).      

Some problems in the theory of differential pursuit games in systems with distributed parameters

A game with dynamics described by partial differential equations is considered. The equations of the players are additively represented on the right-hand side and are subject to integral or pointwise restrictions. The goal of the first player, who is informed of the instantaneous value of the control of his partner, is to bring the system into an unperturbed state. To solve the problem the decomposition method developed in [1] for a controlled system (with one player) is used. Three combinations of restrictions on the players are considered. In all cases the control of the first player is presented explicitly. The main complication, compared with the problem considered previously [1] is that this control consists of two terms estimated in different norms.     

Games of timing with resources of mixed type

The paper considers a game of timing which is closely related to the so-called duels. This is a game connected with the distribution of resources by two players. Each of the players is in possession of some amount of resource to be distributed by him in the time interval [0, 1]. In his behavior, Player 1 is restricted by the necessity of taking all of his resources at a single point, while Player 2 has no restrictions. For the payoff function, defined as for duels, the game is solved; explicit formulas on the value of the game and the optimal strategies for the players are found.     

Generalized solution concepts in games with possibly unaware players

Most work in game theory assumes that players are perfect reasoners and have common knowledge of all significant aspects of the game. In earlier work (Halpern and Rêgo 2006, arxiv.org/abs/0704.2014), we proposed a framework for representing and analyzing games with possibly unaware players, and suggested a generalization of Nash equilibrium appropriate for games with unaware players that we called generalized Nash equilibrium. Here, we use this framework to analyze other solution concepts that have been considered in the game-theory literature, with a focus on sequential equilibrium.     

Asymptotics of degenerations of mixed Hodge structures

We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate [6] between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric introduced by Hain [12], analogs of the norm estimates of [19] and the asymptotics of the naive limit Hodge filtration considered in [21].     

Old and new moving-knife schemes

In general, moving-knife schemes seem to be easier to come by than pure existence results (like Neyman’s [N] theorem) but harder to come by than discrete algorithms (like the Dubins-Spanier [DS] last-diminisher method). For envy-free allocations for four or more people, however, the order of difficulty might actually be reversed. Neyman’s existence proof (for anyn) goes back to 1946, the discovery of a discrete algorithm for alln ≥ 4 is quite recent [BT1, BT2, BT3], and a moving-knife solution forn = 4 was found only as this article was being prepared (see [BTZ]). We are left with this unanswered question: Is there a moving-knife scheme that yields an envyfree division for five (or more) players?