Final Report Summary - LAG (Learning Across Games)
Our research lies in the area of game theory. We are studying the twin questions of:
a) how agents categorise decision problems (and in particular games) according to their similarity; and
b) how this categorisation affects standard predictions in game theory.
The most important contribution to the literature is that we endogenise the question of categorisation, assuming that categorisations can be learned. Existing literature is characterised by making sometimes ad hoc assumptions on categorisations. Our results have shown that some of the key concepts used in game theory appear to be very fragile once the problem of categorisation is introduced. We have also shown that many recent experimental results have a natural explanation in terms of learned categorisations. The results of the different research articles are summarised.
Learning across games
This paper studies the learning process carried out by two agents who are involved in many games. As distinguishing all games can be too costly (require too much reasoning resources) agents might partition the set of all games into categories. Partitions of higher cardinality are more costly. A process of simultaneous learning of actions and partitions is presented and equilibrium partitions and action choices characterised. Learning across games can destabilise strict Nash equilibria and stabilise equilibria in weakly dominated strategies even for arbitrarily small reasoning costs. The model is also able to explain experimental _findings from the Traveller's dilemma or deviations from subgame perfection in bargaining games.
Learning without counterfactuals (joint with J. Rivas)
In this paper we study learning procedures when counterfactuals (payoffs of not-chosen actions) are not observed. The decision maker reasons in two steps: First, she updates her propensities for each action after every payoff experience, where propensity is defined as how much she prefers each action. Then, she transforms these propensities into choice probabilities. We introduce natural axioms in the way propensities are updated and the way propensities are translated into choice, and study the decision marker's behaviour when such axioms are in place.
An experiment on learning in a multiple games environment (joint with V. Grimm)
We study experimentally how players learn to make decisions if they face many different (normal form) games. Games are generated randomly from a uniform distribution in each of 100 rounds. We find that agents do extrapolate between games but learn to play strategically equivalent games in the same way. If either there are few games or if access to information about the opponents' behaviour is easy (or both) convergence to the unique Nash equilibrium generally occurs. Otherwise this is not the case and play converges to a distribution of actions which is non-Nash but which can be rationalised by theoretical models of categorisation. Estimating different learning models we find that Nash choices are best explained by finer categorisations than non-Nash choices. Furthermore, participants scoring better in the 'Cognitive reflection test' (Frederick, 2005) choose Nash actions more often than other participants.
Extrapolation in games of coordination and dominance solvable games (joint with E. Sciubba)
We study extrapolation between games in a laboratory experiment. Participants in our experiment first play either the dominance solvable guessing game or a coordination version of the guessing game for five rounds. Afterwards they play a 3x3 normal form game for ten rounds with random matching which is either a game solvable through Iterated elimination of dominated strategies (IEDS), a pure coordination game or a coordination game with pareto ranked equilibria. We found strong evidence that participants do extrapolate between games.
a) how agents categorise decision problems (and in particular games) according to their similarity; and
b) how this categorisation affects standard predictions in game theory.
The most important contribution to the literature is that we endogenise the question of categorisation, assuming that categorisations can be learned. Existing literature is characterised by making sometimes ad hoc assumptions on categorisations. Our results have shown that some of the key concepts used in game theory appear to be very fragile once the problem of categorisation is introduced. We have also shown that many recent experimental results have a natural explanation in terms of learned categorisations. The results of the different research articles are summarised.
Learning across games
This paper studies the learning process carried out by two agents who are involved in many games. As distinguishing all games can be too costly (require too much reasoning resources) agents might partition the set of all games into categories. Partitions of higher cardinality are more costly. A process of simultaneous learning of actions and partitions is presented and equilibrium partitions and action choices characterised. Learning across games can destabilise strict Nash equilibria and stabilise equilibria in weakly dominated strategies even for arbitrarily small reasoning costs. The model is also able to explain experimental _findings from the Traveller's dilemma or deviations from subgame perfection in bargaining games.
Learning without counterfactuals (joint with J. Rivas)
In this paper we study learning procedures when counterfactuals (payoffs of not-chosen actions) are not observed. The decision maker reasons in two steps: First, she updates her propensities for each action after every payoff experience, where propensity is defined as how much she prefers each action. Then, she transforms these propensities into choice probabilities. We introduce natural axioms in the way propensities are updated and the way propensities are translated into choice, and study the decision marker's behaviour when such axioms are in place.
An experiment on learning in a multiple games environment (joint with V. Grimm)
We study experimentally how players learn to make decisions if they face many different (normal form) games. Games are generated randomly from a uniform distribution in each of 100 rounds. We find that agents do extrapolate between games but learn to play strategically equivalent games in the same way. If either there are few games or if access to information about the opponents' behaviour is easy (or both) convergence to the unique Nash equilibrium generally occurs. Otherwise this is not the case and play converges to a distribution of actions which is non-Nash but which can be rationalised by theoretical models of categorisation. Estimating different learning models we find that Nash choices are best explained by finer categorisations than non-Nash choices. Furthermore, participants scoring better in the 'Cognitive reflection test' (Frederick, 2005) choose Nash actions more often than other participants.
Extrapolation in games of coordination and dominance solvable games (joint with E. Sciubba)
We study extrapolation between games in a laboratory experiment. Participants in our experiment first play either the dominance solvable guessing game or a coordination version of the guessing game for five rounds. Afterwards they play a 3x3 normal form game for ten rounds with random matching which is either a game solvable through Iterated elimination of dominated strategies (IEDS), a pure coordination game or a coordination game with pareto ranked equilibria. We found strong evidence that participants do extrapolate between games.