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Title
Abstract
Figures
Introduction
Related Literature
Results for the Poisson CH Model
Conclusions
References

Cognitive Hierarchies in Extensive Form Games

Profile image of Po-Hsuan LinPo-Hsuan Lin

2022

Abstract

The cognitive hierarchy (CH) approach posits that players in a game are heterogeneous with respect to levels of strategic sophistication. A level-k player believes all other players in the game have lower levels of sophistication distributed from 0 to k 1, and these beliefs correspond to the truncated distribution of a “true” distribution of levels. We extend the CH framework to extensive form games, where these initial beliefs over lower levels are updated as the history of play in the game unfolds, providing information to players about other players’ levels of sophistication. For a class of centipede games with a linearly increasing pie, we fully characterize the dynamic CH solution and show that it leads to the game terminating earlier than in the static CH solution for the centipede game in reduced normal form. JEL Classification Numbers: C72

Figures (10)
"We assume (as is typical in level-k models) that players randomize uniformly over optimal actions when indifferent. This assumption is convenient because it ensures a unique dynamic CH solution to every game, so we assume it here. Note that while the dynamic CH solution is defined as a fixed point, it can be solved for recursively, starting with the lowest level and iteratively working up to higher levels.
"We assume (as is typical in level-k models) that players randomize uniformly over optimal actions when indifferent. This assumption is convenient because it ensures a unique dynamic CH solution to every game, so we assume it here. Note that while the dynamic CH solution is defined as a fixed point, it can be solved for recursively, starting with the lowest level and iteratively working up to higher levels.
Figure 1: Game Tree of Example 1. A ”+” sign indicates a move is chosen by the specified level type and all higher levels. The subgame perfect equilibrium moves are marked with arrows.
Figure 1: Game Tree of Example 1. A ”+” sign indicates a move is chosen by the specified level type and all higher levels. The subgame perfect equilibrium moves are marked with arrows.
Figure 2: Game Tree of Example 2. A ”+” sign indicates a move is chosen by the specified level type and all higher levels. The subgame perfect equilibrium moves are marked with arrows.
Figure 2: Game Tree of Example 2. A ”+” sign indicates a move is chosen by the specified level type and all higher levels. The subgame perfect equilibrium moves are marked with arrows.
Figure 3: Game Tree of Example 3. Dashed lines are the paths selected by level-1 players. Level-1 players believe all other players are level-O. As we compute the expected payoff of each action, level-1 player 1 will choose r at the initial node. Level-1 player 2 will choose
Figure 3: Game Tree of Example 3. Dashed lines are the paths selected by level-1 players. Level-1 players believe all other players are level-O. As we compute the expected payoff of each action, level-1 player 1 will choose r at the initial node. Level-1 player 2 will choose
Table 1: Reduced Normal Form of Example 2 We first revisit Example 2 to show how differently the static CH solution is obtained compared with the logic of the dynamic CH solution. Table 1 displays the 4 x 6 matrix game that is reduced normal form representation of the extensive form game of | Example 2. It is easy to see that level 1 and higher of player 1 will choose the strategy rjql;. and level 1 and higher of player 2 will choose the strategy logl2,, as indicated in the table . Thus, for this example it turns out that both models lead to a solution where behavior corresponds to an equilibrium outcome that differs from the subgame perfect equilibrium outcome.
Table 1: Reduced Normal Form of Example 2 We first revisit Example 2 to show how differently the static CH solution is obtained compared with the logic of the dynamic CH solution. Table 1 displays the 4 x 6 matrix game that is reduced normal form representation of the extensive form game of | Example 2. It is easy to see that level 1 and higher of player 1 will choose the strategy rjql;. and level 1 and higher of player 2 will choose the strategy logl2,, as indicated in the table . Thus, for this example it turns out that both models lead to a solution where behavior corresponds to an equilibrium outcome that differs from the subgame perfect equilibrium outcome.
Figure 4: Game Tree of Example 4. A ”+” sign indicates a move is chosen by the specified level type and all higher levels. The subgame perfect equilibrium moves are marked with arrows. in the dynamic model. Although the payoff 10 is really attractive to player 1, player 1 in the dynamic model will realize he can get it only if player 2 is level-0. Therefore, if there is a high enough probability of higher levels of player 2, player 1 will realize he is likely to get the lower payoff of 3 at node 2c. Hence, a high-level player 1 will choose l;, at the beginning as if conducting backward induction). As long as there are enough strategic types of player 1 choosing [,,, higher levels of player 2 will update accordingly and choose the subgame perfect equilibrium action r2,. The calculations can be found in Appendix A. In contrast, the static CH model makes exactly the same prediction as in Example 2. That is, even though player 1’s subgame equilibrium payoff has increased from 3 to 8, all strategic types of players 1 and 2 in the reduced normal form will still choose rjgl1, and loglap, respectively, again producing the imperfect equilibrium outcome (2, 5).
Figure 4: Game Tree of Example 4. A ”+” sign indicates a move is chosen by the specified level type and all higher levels. The subgame perfect equilibrium moves are marked with arrows. in the dynamic model. Although the payoff 10 is really attractive to player 1, player 1 in the dynamic model will realize he can get it only if player 2 is level-0. Therefore, if there is a high enough probability of higher levels of player 2, player 1 will realize he is likely to get the lower payoff of 3 at node 2c. Hence, a high-level player 1 will choose l;, at the beginning as if conducting backward induction). As long as there are enough strategic types of player 1 choosing [,,, higher levels of player 2 will update accordingly and choose the subgame perfect equilibrium action r2,. The calculations can be found in Appendix A. In contrast, the static CH model makes exactly the same prediction as in Example 2. That is, even though player 1’s subgame equilibrium payoff has increased from 3 to 8, all strategic types of players 1 and 2 in the reduced normal form will still choose rjgl1, and loglap, respectively, again producing the imperfect equilibrium outcome (2, 5).
In this section, we demonstrate the representation effect on the class of “linear centipede umes,” which is illustrated in Figure 5. The games in this class are described in the following ay. Player 1, the first-mover, and player 2, the second-mover, alternate over a sequence moves. At each move, the player whose turn it is can either end the game (“take”) id receive the larger of two payoffs or allow the game to continue (“pass”), in which case rth the large and the small payot! ff are incremented by an amount c > 0. The difference tween the large and the small payoff equals 1 and does not change. The game continues r at most 2S decision nodes (stages) where S > 2, and we label the decision nodes by ,2,...,25}. Player 1 moves at odd nodes and player 2 moves at even nodes. If the game ended by a player at stage 7 < 2S, the payoffs are (1+ (j — 1)c, (j — 1)c) if 7 is odd and j —1)c, 1+ (j —1)c) if 7 is even. If no player ever takes, the payoffs are (1+ 2Sc, 2Sc). hus, a linear centipede game has two parameters: (S,c). To avoid trivial cases, we assume gee {*
In this section, we demonstrate the representation effect on the class of “linear centipede umes,” which is illustrated in Figure 5. The games in this class are described in the following ay. Player 1, the first-mover, and player 2, the second-mover, alternate over a sequence moves. At each move, the player whose turn it is can either end the game (“take”) id receive the larger of two payoffs or allow the game to continue (“pass”), in which case rth the large and the small payot! ff are incremented by an amount c > 0. The difference tween the large and the small payoff equals 1 and does not change. The game continues r at most 2S decision nodes (stages) where S > 2, and we label the decision nodes by ,2,...,25}. Player 1 moves at odd nodes and player 2 moves at even nodes. If the game ended by a player at stage 7 < 2S, the payoffs are (1+ (j — 1)c, (j — 1)c) if 7 is odd and j —1)c, 1+ (j —1)c) if 7 is even. If no player ever takes, the payoffs are (1+ 2Sc, 2Sc). hus, a linear centipede game has two parameters: (S,c). To avoid trivial cases, we assume gee {*
Figure 6: (Left) The minimum value of \ to support taking at stage 2S — 1 for both the dynamic CH model (solid) and the static CH model (dash) when c = 0.8 with S on the horizontal axis and A on the vertical axis. (Right) The CDF of terminal nodes in four-node centipede games predicted by the dynamic CH and static CH models.
Figure 6: (Left) The minimum value of \ to support taking at stage 2S — 1 for both the dynamic CH model (solid) and the static CH model (dash) when c = 0.8 with S on the horizontal axis and A on the vertical axis. (Right) The CDF of terminal nodes in four-node centipede games predicted by the dynamic CH and static CH models.
As the prior distribution follows Poisson(A), the condition becomes
As the prior distribution follows Poisson(A), the condition becomes

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