Computational Extensive-Form Games

We define solution concepts appropriate for computationally bounded players playing a fixed finite game. To do so, we need to define what it means for a \emph{computational game}, which is a sequence of games that get larger in some appropriate sense, to represent a single finite underlying extensive-form game. Roughly speaking, we require all the games in the sequence to have essentially the same structure as the underlying game, except that two histories that are indistinguishable (i.e., in the same information set) in the underlying game may correspond to histories that are only computationally indistinguishable in the computational game. We define a computational version of both Nash equilibrium and sequential equilibrium for computational games, and show that every Nash (resp., sequential) equilibrium in the underlying game corresponds to a computational Nash (resp., sequential) equilibrium in the computational game. One advantage of our approach is that if a cryptographic protocol represents an abstract game, then we can analyze its strategic behavior in the abstract game, and thus separate the cryptographic analysis of the protocol from the strategic analysis

Extensive-form games and strategic complementarities

I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a nonempty, complete lattice—in particular, subgame-perfect Nash equilibria exist. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. My results are limited because extensive-form games of strategic complementarities turn out—surprisingly—to be a very restrictive class of games

Extensive-form games and strategic complementarities

(less than 25 lines) I prove the subgame-perfect equivalent of the basic result for Nash equilibria in normal-form games of strategic complements: the set of subgame-perfect equilibria is a non-empty, complete lattice. For this purpose I introduce a device that allows the study of the set of subgame-perfect equilibria as the set of fixed points of a correspondence. The correspondence has a natural interpretation. My results are limited because extensive-form games of strategic complementarities turn out---surprisingly---to be a very restrictive class of games.

Selling information in extensive form games.

We consider a situation in wich decision markets in an extensive form game can by aditional information from an information seller before reaching their decisions. Prices for information are selected by the seller. We analyze a variety of scenarios for the price sitting process by the seller: the case in which prices are chosen before the game stars (ex-ante pricing), the case in which prices are chosen during the game (ex-post pricing) and the situation in which the seller can pit buyers against each other in determining what information is to be sold. Within the context of ex-ante pricin, we also consider the situaction in which the pricise information offered to the decision makers is not exogenously given but is selected by the seller.Extensive form games; Information;

Quantum Extensive Form Games

We propose a concept of quantum extensive-form games, which is a quantum extension of classical extensive-form games. Extensive-form games is a general concept of games such as Go, Shogi, and chess, which have triggered the recent AI revolution, and is the basis for many important game theoretic models in economics. Quantum transitions allow for pairwise annihilation of paths in the quantum game tree, resulting in a probability distribution that is more likely to produce a particular outcome. This is similar in principle to the mechanism of speed-up by quantum computation represented by Grover's algorithm. A quantum extensive-form game is also a generalization of quantum learning, including Quantum Generative Adversarial Networks. Therefore it will become new theoretic basis of quantum machine learning, as well as a basis for a new game theoretic foundation for microeconomics. We propose the quantum angel problem as a new example of quantum extensive-form games. This is a quantum version of angel problem proposed by Conway in 1996. His original problem has already been solved, but by quantizing it, it becomes a non-trivial problem. In the quantum angel problem, Angel moves on a general graph as a quantum walker. By not only changing the dimensions and geometry of the graph, but also by adding/relaxing restrictions to the quantum resources available to Angel and Devil, the difficulty and complexity of the game is diversified in a way that is not possible in the traditional angel problem.Comment: 12 page

Prudent Rationalizability in Generalized Extensive-Form Games

We define an extensive-form analogue of iterated admissibility, called Prudent Rationalizability (PR). In each round of the procedure, for each information set of a player a surviving strategy of hers is required to be rational vis-a-vis a belief system with a full-support belief on the opponents' previously surviving strategies that reach that information set. Somewhat surprisingly, prudent rationalizable strategies may not refine the set of Extensive-Form Rationalizable (EFR) strategies (Pearce 1984). However, we prove that the paths induced by PR strategy-profiles (weakly) refine the set of paths induced by EFR strategies. PR applies also to generalized extensive-form games which model mutual unawareness of actions (Heifetz, Meier and Schipper, 2011a). We demonstrate the applicability of PR in the analysis of verifiable communication, and show that it yields the same, full information unraveling prediction as does the unique sequential equilibrium singled out by Milgrom and Roberts (1986); yet, we also show that under unawareness full unraveling might fail.Prudent rationalizability, caution, extensive-form rationalizability, extensive-form games, unawareness, verifiable communication

Solving Large Extensive-Form Games with Strategy Constraints

Extensive-form games are a common model for multiagent interactions with imperfect information. In two-player zero-sum games, the typical solution concept is a Nash equilibrium over the unconstrained strategy set for each player. In many situations, however, we would like to constrain the set of possible strategies. For example, constraints are a natural way to model limited resources, risk mitigation, safety, consistency with past observations of behavior, or other secondary objectives for an agent. In small games, optimal strategies under linear constraints can be found by solving a linear program; however, state-of-the-art algorithms for solving large games cannot handle general constraints. In this work we introduce a generalized form of Counterfactual Regret Minimization that provably finds optimal strategies under any feasible set of convex constraints. We demonstrate the effectiveness of our algorithm for finding strategies that mitigate risk in security games, and for opponent modeling in poker games when given only partial observations of private information.Comment: Appeared in AAAI 201

Causal assessment in finite extensive-form games

Two finite extensive-form games are empirically equivalent when the empirical distribution on action profiles generated by every behavior strategy in one can also be generated by an appropriately chosen behavior strategy in the other. This paper provides a characterization of empirical equivalence. The central idea is to relate a game's information structure to the conditional independencies in the empirical distributions it generates. We present a new analytical device, the influence opportunity diagram of a game, describe how such a diagram is constructed for a given extensive-form game, and demonstrate that it provides a complete summary of the information needed to test empirical equivalence between two games.Causality, structural uncertainty, extensive form games