scholarly journals Generalized Backward Induction: Justification for a Folk Algorithm

Games ◽  
2019 ◽  
Vol 10 (3) ◽  
pp. 34
Author(s):  
Marek Mikolaj Kaminski
Keyword(s):  
Imperfect Information ◽  
Infinite Horizon ◽  
Backward Induction ◽  
Behavioral Strategies ◽  
Infinite Games ◽  
Sequential Games ◽  
Finite Games ◽  
Pure Strategies ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

I introduce axiomatically infinite sequential games that extend Kuhn’s classical framework. Infinite games allow for (a) imperfect information, (b) an infinite horizon, and (c) infinite action sets. A generalized backward induction (GBI) procedure is defined for all such games over the roots of subgames. A strategy profile that survives backward pruning is called a backward induction solution (BIS). The main result of this paper finds that, similar to finite games of perfect information, the sets of BIS and subgame perfect equilibria (SPE) coincide for both pure strategies and for behavioral strategies that satisfy the conditions of finite support and finite crossing. Additionally, I discuss five examples of well-known games and political economy models that can be solved with GBI but not classic backward induction (BI). The contributions of this paper include (a) the axiomatization of a class of infinite games, (b) the extension of backward induction to infinite games, and (c) the proof that BIS and SPEs are identical for infinite games.

scholarly journals Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting

2015 ◽  
Vol 27 (5) ◽  
pp. 751-761 ◽  
Author(s):  
SAMSON ABRAMSKY ◽  
VIKTOR WINSCHEL
Keyword(s):  
Game Theory ◽  
Infinite Horizon ◽  
Infinite Games ◽  
Adequate Treatment ◽  
Infinite Systems ◽  
Deviation Principle ◽  
Game Trees ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria ◽  
Extensive Games

We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations. Certain strategy profiles are proved to be subgame-perfect equilibria using a novel proof principle of predicate coinduction which is shown to be sound. We characterize all subgame-perfect equilibria for the dollar auction game. The economically interesting feature is that in order to prove these results we do not need to rely on continuity assumptions on the pay-offs which amount to discounting the future. In particular, we prove a form of one-deviation principle without any such assumptions. This suggests that coalgebra supports a more adequate treatment of infinite-horizon models in game theory and economics.

scholarly journals Selection functions, bar recursion and backward induction

2010 ◽  
Vol 20 (2) ◽  
pp. 127-168 ◽  
Author(s):  
MARTÍN ESCARDÓ ◽  
PAULO OLIVA
Keyword(s):  
Proof Theory ◽  
Optimal Strategies ◽  
Backward Induction ◽  
Constructive Mathematics ◽  
Infinite Games ◽  
Sequential Games ◽  
Finite Games ◽  
Conceptual Explanation ◽  
Cartesian Closed ◽  
Countably Infinite

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

scholarly journals Dynamic games with (almost) perfect information

2020 ◽  
Vol 15 (2) ◽  
pp. 811-859
Author(s):  
Wei He ◽  
Yeneng Sun
Keyword(s):  
Dynamic Games ◽  
Pure Strategy ◽  
Infinite Horizon ◽  
Complete Information ◽  
Perfect Information ◽  
State Variables ◽  
Deterministic Dynamic ◽  
Upper Hemicontinuity ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

This paper aims to solve two fundamental problems on finite‐ or infinite‐horizon dynamic games with complete information. Under some mild conditions, we prove the existence of subgame‐perfect equilibria and the upper hemicontinuity of equilibrium payoffs in general dynamic games with simultaneous moves (i.e., almost perfect information), which go beyond previous works in the sense that stagewise public randomization and the continuity requirement on the state variables are not needed. For alternating move (i.e., perfect‐information) dynamic games with uncertainty, we show the existence of pure‐strategy subgame‐perfect equilibria as well as the upper hemicontinuity of equilibrium payoffs, extending the earlier results on perfect‐information deterministic dynamic games.

scholarly journals Search for a moving target in a competitive environment

2021 ◽  
Author(s):  
Benoit Duvocelle ◽  
János Flesch ◽  
Hui Min Shi ◽  
Dries Vermeulen
Keyword(s):  
Markov Chain ◽  
Discrete Time ◽  
Competitive Environment ◽  
Moving Target ◽  
Time Varying ◽  
Greedy Strategy ◽  
Time Dynamic ◽  
Dynamic Search ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

AbstractWe consider a discrete-time dynamic search game in which a number of players compete to find an invisible object that is moving according to a time-varying Markov chain. We examine the subgame perfect equilibria of these games. The main result of the paper is that the set of subgame perfect equilibria is exactly the set of greedy strategy profiles, i.e. those strategy profiles in which the players always choose an action that maximizes their probability of immediately finding the object. We discuss various variations and extensions of the model.

Interpreting Knowledge in the Backward Induction Problem

Episteme ◽  
2011 ◽  
Vol 8 (3) ◽  
pp. 248-261 ◽  
Author(s):  
Ken Binmore
Keyword(s):  
Common Knowledge ◽  
Perfect Information ◽  
Backward Induction ◽  
Finite Games ◽  
Common Knowledge Of Rationality

AbstractRobert Aumann argues that common knowledge of rationality implies backward induction in finite games of perfect information. I have argued that it does not. A literature now exists in which various formal arguments are offered in support of both positions. This paper argues that Aumann's claim can be justified if knowledge is suitably reinterpreted.

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