Noncooperative Games In Normal Form Research Articles

Perfect Prediction in normal form: Superrational thinking extended to non-symmetric games

This paper introduces a new solution concept for non-cooperative games in normal form with no ties and pure strategies: the Perfectly Transparent Equilibrium. The players are rational in all possible worlds and know each other’s strategies in all possible worlds — which, together, we refer to as Perfect Prediction. The anticipation of a player’s decision by their opponents is counterfactually dependent on the decision, unlike in Nash Equilibria where the decisions are made independently. The equilibrium, when it exists, is unique and is Pareto optimal.This equilibrium is the normal-form counterpart of the Perfect Prediction Equilibrium; the prediction happens “in another room” rather than in the past. The equilibrium can also be seen as a natural extension of Hofstadter’s superrationality to non-symmetric games. Algorithmically, an iterated elimination of non-individually-rational strategy profiles is performed until at most one remains. An equilibrium is a strategy profile that is immune against knowledge of strategies in all possible worlds and rationality in all possible worlds, a stronger concept than common knowledge of rationality but also stronger than common counterfactual belief of rationality.We formalize and contrast the Non-Nashian Decision Theory paradigm, common to this and several other papers, with Causal Decision Theory and Evidential Decision Theory. We define the Perfectly Transparent Equilibrium algorithmically and prove (when it exists) that it is unique, that it is Pareto-optimal, and that it coincides with Hofstadter’s Superrationality on symmetric games. We relate the finding to concepts found in the literature such as Individual Rationality, Rationalizability, Minimax-Rationalizability, Second-Order Nash Equilibria, the Program Equilibrium, the Perfect Prediction Equilibrium, Shiffrin’s Joint-Selfish-Rational Equilibrium, the Stalnaker–Bonanno Equilibrium, the Perfect Cooperation Equilibrium, the Translucent Equilibrium, the Correlated Equilibrium, and Quantum Games. Finally, we specifically discuss inclusion relationships on the special case of symmetric games between Individual Rationality, Minimax-Rationalizability, Superrationality, and the Perfectly Transparent Equilibrium, and contrast them with asymmetric games.

Non-cooperative games with prospect theory players and dominated strategies

We investigate a framework for non-cooperative games in normal form where players have behavioral preferences following Prospect Theory (PT) or Cumulative Prospect Theory (CPT). On theoretical grounds CPT is usually considered to be the superior model, since it normally does not violate first order stochastic dominance in lottery choices. We find, however, that CPT when applied to games may select purely dominated strategies, while PT does not. For both models we also characterize the cases where mixed dominated strategies are preserved and where violations may occur.

The Computational Complexity of Finding a Mixed Berge Equilibrium for a k-Person Noncooperative Game in Normal Form

The mixed Berge equilibrium (MBE) is an extension of the Berge equilibrium (BE) to mixed strategies. The MBE models mutually support in a [Formula: see text]-person noncooperative game in normal form. An MBE is a mixed-strategy profile for the [Formula: see text] players in which every [Formula: see text] players have mixed strategies that maximize the expected payoff for the remaining player’s equilibrium strategy. In this paper, we study the computational complexity of determining whether an MBE exists in a [Formula: see text]-person normal-form game. For a two-person game, an MBE always exists and the problem of finding an MBE is PPAD-complete. In contrast to the mixed Nash equilibrium, the MBE is not guaranteed to exist in games with three or more players. Here we prove, when [Formula: see text], that the decision problem of asking whether an MBE exists for a [Formula: see text]-person normal-form game is NP-complete. Therefore, in the worst-case scenario there does not exist a polynomial algorithm that finds an MBE unless P=NP.

Deriving Nash equilibria as the supercore for a relational system

Abstract In this paper, under a binary relation that refines the standard relation which only accounts for single profitable deviations, we obtain that the set of NE strategy profiles of every finite non-cooperative game in normal form coincides with the supercore ( Roth, 1976 ) of its associated abstract system. Further, under the standard relation we show when these two solution concepts coincide.

Nash Equilibria in Pure Strategies

We consider an n‐person non‐zero‐sum non‐cooperative game in normal form, where the strategy sets are some closed intervals of the real line. It is shown that if the pay‐off functions are continuous on the whole space and if for each pay‐off function the smallest local maximum in the strategy variable is a global maximum, then the game possesses a pure strategy Nash equilibrium.

Strongly Stable Equilibrium Points of N-Person Noncooperative Games

We introduce a concept of a strongly stable (Nash) equilibrium point of an n-person noncooperative game in normal form. Roughly speaking, an equilibrium point is strongly stable if it changes continuously and uniquely against any small perturbation to payoffs of players. We establish a necessary and sufficient condition for an equilibrium point to be strongly stable. By the Karush-Kuhn-Tuckcr optimality condition for nonlinear programs, an equilibrium point is shown to correspond to a zero-point of a certain PC1-mapping. Then our condition is stated in terms of the local nonsingularity of the Jacobian matrix of this mapping. Our proof is based on the degree theory of mappings. Finally, we give an example of a 3-person game with two nonquasi-strong equilibrium points, only one of which is strongly stable.

A Core Existence Theorem for Games Without Ordered Preferences

[Introduction] To a large extent the cooperative theory of games has an altogether different appearance from the noncooperative theory. The noncooperative theory generally deals with games in either extensive form or normal form, while the cooperative theory is usually described in characteristic function form. One of the central concepts in the cooperative theory is that of the core, which is the set of utility allocations which no coalition can improve upon. This notion of the core and of the characteristic function form of a game depends heavily on the existence of a utility representation for players' preferences. Recently Gale and Mas-Colell [3] and Shafer and Sonnenschein [6] have proven theorems on the existence of a Nash equilibrium for noncooperative games in normal form in which the players' preferences over strategy vectors are not necessarily complete or transitive and so may fail to have a utility representation. Thus it might appear that the noncooperative theory is applicable in environments where the cooperative theory is not. In order to formulate theorems in the cooperative theory of games which can be applied to environments in which players may have nonordered preferences, the characteristic function must be reformulated in terms of physical outcomes as opposed to utility outcomes. The players' preferences can then be expressed in terms of the physical outcomes without the use of a utility function.

Equilibrium points of nonatomic games

The Nash theorem on the existence of equilibrium points inTV-person non-cooperative games in normal form is generalized to the case when there is a continuum of players endowed with a nonatomic measure. The mathematical tools are those used in mathematical economics, in particular, markets with a continuum of traders. The main result shows that under a restriction on the payoff functions there exists an equilibrium in pure strategies.

Computing Equilibria of Two-Person Games from the Extensive Form

The Lemke-Howson algorithm for computing equilibria of finite 2-person non-cooperative games in normal form is modified to restrict the computations to the ordinarily small portion corresponding to the strategies actually used by the players, and further it is shown that in games with perfect recall these strategies can be generated as needed from an auxiliary analysis of the players' decision trees derived from the extensive form of the game.

Cores in noncooperative games

Noncooperative games in normal form and in characteristic function form are considered. The supergame of the noncooperative game is defined as an infinite sequence of plays of the original game. The notions of strong Pareto equilibrium point (s.p.e.p.) and essential core are introduced. A relationship between the essential core of a noncooperative game and the set of s.p.e.p. of its supergame is asserted. This result is similar to that ofAumann for cooperative games without side payments.

Some Recent Developments inn-Person Game Theory

A general survey of the mathematical theory of games is presented with particular emphasis on recent developments in the theory of the core and stable sets (solutions) for the multi-person cooperative games in characteristic function form. Extensions to games without side payments, games with infinitely many players, and games in partition function form are mentioned. The history leading up to the discovery of a ten-person game with no stable set is reviewed. The equilibrium theory for noncooperative games in normal form as well as some additional cooperative solution concepts are covered very briefly.