Recursive Reasoning by Humans in Strategic Games

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What is Recursive Reasoning by Humans in Strategic Games?
First-level reasoning and second-level reasoning in a two-player alternating-move centipede game.

Recursive reasoning of the form what do I think that you think that I think (and so on) arises often while acting rationally in multi-agent settings. When humans play in strategic games, in order to maximize their payoffs, they might also need to consider about their opponents' strategies which in turn may depend on their strategies (and so on).

Previous studies show that humans generally operate at only first or second level of recursive reasoning. Typically the first level, which attributes no recursive reasoning to others, is more prominent. Evidence of these shallow levels of reasoning is not surprising, as humans are limited by bounded rationality. However, in our studies, we find out that in some games settings, humans can display higher levels of reasoning.

Project Description
(a) A game tree representation (extensive form) of our two-player game. Because of its particular structure, such games are also sometimes called Centipede games. States of the game are indicated by the letters, A, B, C and D. (b) Arrows denote the progression of play in the game. An action of move by each player causes a transition of the state of the game.

We designed experiments to study the levels of recursive reasoning displayed by humans in strategic games. The game we selected is a two-player alternating-move game of complete and perfect information, as seen in figure.

Player I (the leader) may elect to move or stay. If player I elects to move, player II (the follower) faces the choice of moving or staying, as well. An action of stay by any player terminates the game. Note that actions of all players are perfectly observable to each other and payoffs for both players at different possible stop ends are also observable. The game may be extended to any number of moves. Two studies are conducted based on the length of games, which also decides the highest possible level of reasoning that one player might have.

ToM2: Game was terminated after two moves of player I, as seen in figure. In this study, the highest possible level of reasoning is the second level. In this study, two experiments are conducted.

Experiment 1 (General-Sum Game): Payoffs for both players at each possible stop end have no direct relation, as seen in figure.

Experiment 2 (Fixed-Sum Game): The sum of the payoffs for both players at each possible stop end is fixed, as seen in figure.

A four-stage fixed-sum sequential game in (a) matrix and (b) tree format.

ToM3: Game was terminated after two moves of player II, as seen in figure. In this study, the highest possible level of reasoning is the third level.

Computational models of the behavioral data are being obtained from the studies using the I-POMDP framework, appropriately augmented using well-known human judgement and decision models.

Project Generated Resources

Papers:

  1. Adam Goodie, Prashant Doshi and Diana Young, "Two Level Recursive Reasoning by Humans Playing Sequential Fixed-Sum Games", in MSDM Workshop, 2009
  2. Prashant Doshi, Xia Qu, Adam Goodie and Diana Young, "Modeling Recursive Reasoning in Humans Using Empirically Informed Interactive POMDPs", in AAMAS 2010
  3. Adam Goodie, Prashant Doshi and Diana Young, "Levels of Theory-of-Mind Reasoning in Competitive Games", in the Journal of Behavioral Decision Making (BDM), Wiley, 2010
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