Rights statement: © 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
Final published version, 647 KB, PDF document
Available under license: CC BY
Final published version
Licence: CC BY
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - Stochastic fictitious play with continuous action sets
AU - Perkins, S.
AU - Leslie, D. S.
N1 - © 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
PY - 2014/7
Y1 - 2014/7
N2 - Continuous action space games are ubiquitous in economics. However, whilst learning dynamics in normal form games with finite action sets are now well studied, it is not until recently that their continuous action space counterparts have been examined. We extend stochastic fictitious play to the continuous action space framework. In normal form games with finite action sets the limiting behaviour of a discrete time learning process is often studied using its continuous time counterpart via stochastic approximation. In this paper we study stochastic fictitious play in games with continuous action spaces using the same method. This requires the asymptotic pseudo-trajectory approach to stochastic approximation to be extended to Banach spaces. In particular the limiting behaviour of stochastic fictitious play is studied using the associated smooth best response dynamics on the space of finite signed measures. Using this approach, stochastic fictitious play is shown to converge to an equilibrium point in two-player zero-sum games and a stochastic fictitious play-like process is shown to converge to an equilibrium in negative definite single population games.
AB - Continuous action space games are ubiquitous in economics. However, whilst learning dynamics in normal form games with finite action sets are now well studied, it is not until recently that their continuous action space counterparts have been examined. We extend stochastic fictitious play to the continuous action space framework. In normal form games with finite action sets the limiting behaviour of a discrete time learning process is often studied using its continuous time counterpart via stochastic approximation. In this paper we study stochastic fictitious play in games with continuous action spaces using the same method. This requires the asymptotic pseudo-trajectory approach to stochastic approximation to be extended to Banach spaces. In particular the limiting behaviour of stochastic fictitious play is studied using the associated smooth best response dynamics on the space of finite signed measures. Using this approach, stochastic fictitious play is shown to converge to an equilibrium point in two-player zero-sum games and a stochastic fictitious play-like process is shown to converge to an equilibrium in negative definite single population games.
KW - Stochastic fictitious play
KW - Learning in games
KW - Continuous action set games
KW - Abstract stochastic approximation
U2 - 10.1016/j.jet.2014.04.008
DO - 10.1016/j.jet.2014.04.008
M3 - Journal article
VL - 152
SP - 179
EP - 213
JO - Journal of Economic Theory
JF - Journal of Economic Theory
SN - 0022-0531
ER -