Further Results on Verification Problems in Extensive-Form Games

49 Pages Posted: 16 Jul 2016

See all articles by Nicola Gatti

Nicola Gatti

Polytechnic University of Milan

Mario Gilli

Università degli Studi di Milano-Bicocca - Center for Interdisciplinary Studies in Economics, Psychology & Social Sciences (CISEPS); Università degli Studi di Milano-Bicocca - Department of Economics, Management and Statistics (DEMS)

Fabio Panozzo

Polytechnic University of Milan

Date Written: July 15, 2016

Abstract

The computational study of games is receiving increasing attention both in game theory and computer science. The challenge is distinguishing computationally tractable problems (also said easy), admitting polynomial–time algorithms, from the intractable ones (also said hard). In this paper, we focus on extensive form games, as the computational problems defined on such games are largely unexplored. We study the problem (aka verification problem) of certifying that a solution given in input is an equilibrium according to different refinements for extensive form games as the input change. We show that, when the input is a realization plan strategy profi (i.e., strategies for the sequence form representation), deciding whether the input is a Subgame Perfect Equilibrium or is a part of a Sequential Equilibrium is NP-hard even in two-player games (we conjecture the same holds also for Quasi Perfect Equilibrium). This means that there is no polynomial–time algorithm unless P = NP, but it is commonly believed that P ≠ NP. Subsequently, we show that in two-player games, when the input is a behavioral strategy profile there is a polynomial-time algorithm deciding whether the input is a Quasi-Perfect Equilibrium, and a simple variation of the algorithm decides whether the input is part of some Sequential Equilibrium (in games with three or more players, the problem is known to be NP-hard for both Quasi-Perfect Equilibrium and Sequential Equilibrium). Finally, we show that, when the input is an assessment, there is a polynomial-time algorithm deciding whether the input is a Sequential Equilibrium regardless the number of players.

Keywords: Efficient algorithms, extensive–form refinements

Suggested Citation

Gatti, Nicola and Gilli, Mario and Panozzo, Fabio, Further Results on Verification Problems in Extensive-Form Games (July 15, 2016). University of Milan Bicocca Department of Economics, Management and Statistics Working Paper No. 347, Available at SSRN: https://ssrn.com/abstract=2810056 or http://dx.doi.org/10.2139/ssrn.2810056

Nicola Gatti

Polytechnic University of Milan ( email )

Piazza Leonardo da Vinci
Milan, Milano 20100
Italy

Mario Gilli (Contact Author)

Università degli Studi di Milano-Bicocca - Center for Interdisciplinary Studies in Economics, Psychology & Social Sciences (CISEPS) ( email )

Piazza dell'Ateneo Nuovo, 1
Milano, 20126
Italy

Università degli Studi di Milano-Bicocca - Department of Economics, Management and Statistics (DEMS) ( email )

Piazza dell'Ateneo Nuovo, 1
Milan, 20126
Italy

Fabio Panozzo

Polytechnic University of Milan ( email )

Piazza Leonardo da Vinci
Milan, Milano 20100
Italy

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