Experimental Design
This experiment will be conducted at the SSEL at the California Institute of Technology and the ESSL at UC Irvine. In the following sections, we will provide detailed descriptions of the selection of centipede games, the difference between the three representations, and the implementation of the experiment.
Selection of Centipede Games
The representation effect predicted by the dynamic cognitive hierarchy solution is robust across different classes of centipede games. To explore the robustness of this theoretical prediction, we consider three parameterized classes of six-move centipede games: linear centipede games, exponential centipede games, and constant sum centipede games. See the analysis plan for the game trees.
Each class of centipede games can be parameterized by a single parameter. The class of linear centipede games can be parameterized by setting the payoffs of the first and second movers to 1 and 0, respectively, when the game ends at the first node. Payoffs then increase by an amount of c>0 when the game proceeds to the next node. Similarly, the class of exponential centipede games can be parameterized by setting the payoffs of the first and second movers to p>1 and 1, respectively, when the game ends at the first node. Payoffs double when the game proceeds to the next node. Finally, the class of constant sum centipede games can be parameterized by setting the payoffs of both players to be 1 when the game ends at the first node. The smaller payoff is then multiplied by d<1 when the game proceeds to the next node, while the total payoffs remain at 2.
The selection of payoff parameters follows the optimal design approach as outlined by Lin (2023), comprising two main steps: calibration and payoff selection. Initially, we calibrate the distribution of levels by estimating the Poisson-dynamic cognitive hierarchy solution using pilot data. Subsequently, we treat this estimated distribution as the true distribution of levels and select payoff parameters to maximize the expected magnitude of the representation effect, as predicted by the dynamic cognitive hierarchy solution. Specifically, these parameters correspond to the following games: a linear game with c=0.5, an exponential game with p=2.5, and a constant sum game with d=0.8. Additionally, to establish a benchmark, we include three centipede games anticipated to yield a smaller representation effect. These consist of a linear game with c=0.8, an exponential game with p=4, and a constant sum game with d=0.4.
Extensive form, Reduced normal form, and Non-reduced normal form
Extensive form representation: When the game is played according to the extensive form representation, subjects can only make decisions when the game reaches their decision nodes. Under this representation, the centipede game begins at the first decision node, where the first mover decides to either take or pass. If the first mover takes, the game ends. Otherwise, it is the second mover’s turn to decide whether to take or pass. If the second mover passes, it is again the first mover’s turn to decide, and so on. Notice that when the game is played according to the extensive form, subjects will mechanically see their opponent’s previous decisions.
Reduced normal form representation: When the game is played according to the reduced normal form representation, subjects are asked to “simultaneously” choose a reduced contingent strategy, and the payoffs are realized as if the game were played under the extensive form. In the context of six-move centipede games, both players will simultaneously choose one of the four stopping strategies: taking at the first, second, or third chance, or always passing. The payoffs are determined by the player who takes earlier. It is worth noting that under this representation, subjects will not see their opponent’s actual decisions while making their own.
Non-reduced normal form representation: When the game is played according to the non-reduced normal form representations, subjects are asked to simultaneously choose a non-reduced contingent strategy at all their own decision nodes, regardless of whether the node is on-path or off-path. In the context of six-move centipede games, both players will simultaneously choose a contingent strategy that specifies their actions at all three decision nodes, even if they have chosen to take at an earlier node. The payoffs are determined by the player who takes earlier. Similar to the reduced normal form, under this representation, subjects will not see their opponent’s actual decisions while making their own.
Implementation
At the beginning of the experiment, each subject is randomly assigned to one of the two groups, either the first mover or the second mover. The group assignment remains the same throughout the whole experiment.
The experiment consists of three parts, with each part corresponding to one of the three representations. In each part, subjects will play each of the six centipede games only once. The sequence of the games is randomized in each part. Moreover, in each game, a subject is matched with another subject from the other group who has never been matched with in this part. In other words, in each part of the experiment, a subject will not be matched with the same subject from the other group twice.
To mitigate the impact of feedback from repeated play, we employ the non-reduced normal form and reduced normal form representations in the first two parts (without feedback) and the extensive form representation in the third part. Lastly, to control for order effects, we consider two different orders:
• Order 1: Non-reduced normal form, reduced normal form, extensive form;
• Order 2: Reduced normal form, non-reduced normal form, extensive form.
At the end of the experiment, a game from each part will be randomly selected to be realized. Subjects are paid privately based on the sum of the payoffs from the chosen games.
This experimental design possesses several merits:
1. The experimental design enables us to test the representation effect predicted by the dynamic cognitive hierarchy solution by comparing the behavioral differences under three different representations.
2. This experimental design allows us to isolate the potential order effect between the reduced normal form and non-reduced normal form representations by incorporating two different orders in the first two parts.
3. This experimental design minimizes the confounding effect of learning across representations by placing the non-reduced normal form and the reduced normal form without feedback in the first two parts and placing the extensive form representation in the last part.
4. The implementation of different orders of representations allows us to examine whether the order of reduced and non-reduced normal form will affect the behavior under the extensive form representation.
In summary, these design features provide a comprehensive exploration of the representation effect and the order effect in our experiment.