Study of potential games using Ising interaction

Problems can be handled properly in game theory as long as a countable number of players are considered, whereas, in real life, we have a large number of players. Hence, games at the thermodynamic limit are analyzed in general. There is a one-to-one correspondence between classical games and the modeled Hamiltonian at a particular equilibrium condition, usually the Nash equilibrium. Such a correspondence is arrived for symmetric games, namely the Prisoner’s Dilemma using the Ising Hamiltonian. In this work, we have shown that another class of games known as potential games can be analyzed with the Ising Hamiltonian. Analysis of this work brings out very close observation with real-world scenarios. In other words, the model of a potential game studied using Ising Hamiltonian predicts behavioral aspects of a large population precisely.

Decomposed 2*2 games - a conceptual review

2*2 games, such as the Prisoner's Dilemma, are a common tool for studying cooperation and social decision-making. In experiments, 2*2 games are usually presented in matrix form, such that participants see only the possible outcomes. Some 2*2 games can be decomposed into payoffs for self and other, such that participants see the direct consequences of two actions. While the final outcomes of the decomposed form and the matrix-form can be identical, the framing differs: the matrix form emphasises the outcome, the decomposed form emphasises the action. This allows decomposed games to address questions that could not be answered with matrix games. Here, we provide a conceptual overview of decomposed games that is accessible without knowing the underlying mathematics. We explain which 2*2 games can be decomposed, why the same payoff matrix can be decomposed into infinitely many decompositions, and we apply this to (a)symmetric games, (a)symmetric decompositions, and games with ties. Finally, we show how to calculate all decompositions for a given game and we suggest when the decomposed form might be more appropriate than the matrix form for an experimental design.

Reactive Strategies: An Inch of Memory, a Mile of Equilibria

We explore how an incremental change in complexity of strategies (“an inch of memory”) in repeated interactions influences the sets of Nash Equilibrium (NE) strategy and payoff profiles. For this, we introduce the two most basic setups of repeated games, where players are allowed to use only reactive strategies for which a probability of players’ actions depends only on the opponent’s preceding move. The first game is trivial and inherits equilibria of the stage game since players have only unconditional (memory-less) Reactive Strategies (RSs); in the second one, players also have conditional stochastic RSs. This extension of the strategy sets can be understood as a result of evolution or learning that increases the complexity of strategies. For the game with conditional RSs, we characterize all possible NE profiles in stochastic RSs and find all possible symmetric games admitting these equilibria. By setting the unconditional benchmark as the least symmetric equilibrium payoff profile in memory-less RSs, we demonstrate that for most classes of symmetric stage games, infinitely many equilibria in conditional stochastic RSs (“a mile of equilibria”) Pareto dominate the benchmark. Since there is no folk theorem for RSs, Pareto improvement over the benchmark is the best one can gain with an inch of memory.

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author’s best knowledge, the only numerical method in the literature is the heuristic one we put forward in Aïd et al (ESAIM Proc Surv 65:27–45, 2019) to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory.

Evolutionary Game Theory: A Generalization of the ESS Definition

In this paper, we study the concept of Evolutionarily Stable Strategies (ESSs) for symmetric games with [Formula: see text] players. The main properties of these games and strategies are analyzed and several examples are provided. We relate the concept of ESS with previous literature and provide a proof of finiteness of ESS in the context of symmetric games with [Formula: see text] players. We show that unlike the case of [Formula: see text], when there are more than two populations an ESS does not have a uniform invasion barrier, or equivalently, it is not equivalent to the strategy performing better against all strategies in a neighborhood. We also construct the extended replicator dynamics for these games and we study an application to a model of strategic planning of investment.

Superrational types

We present a formal analysis of Douglas Hofstadter’s concept of superrationality. We start by defining superrationally justifiable actions, and study them in symmetric games. We then model the beliefs of the players, in a way that leads them to different choices than the usual assumption of rationality by restricting the range of conceivable choices. These beliefs are captured in the formal notion of type drawn from epistemic game theory. The theory of coalgebras is used to frame type spaces and to account for the existence of some of them. We find conditions that guarantee superrational outcomes.