Approximate Common Beliefs in Rationality in Dynamic Games

Participants will play a series dynamic games. The study uses a within-subject design where each participant plays a set of two-player games, including a Centipede game, Battle of the Sexes with an outside option and a modified version of the game in Figure 4 on Reny 1992 with the same structure as the Battle of the Sexes with an outside option. The experiment is designed to compare behavior under classic dynamic refinements (Subgame Perfect Equilibrium) versus the predictions of Approximate Equilibrium. In addition to making choices, participants will be asked to report their beliefs about other players' actions, which will be incentivized using a scoring rule. To control for potential confounding factors, we randomize role assignment as well as game order. All interactions are one-shot to avoid repeated-game effects.

2026-06-07

2026-06-20

-- Frequency of choices consistent with AEp vs. SPNE/FI predictions in the Modified Reny Game (the critical test game where both solution concepts yield mutually exclusive behavioral predictions).
-- Frequency of choices consistent with AEp vs. SPNE/FI predictions in the Centipede Game and Battle of the Sexes with outside option (benchmark games).

For each game, each subject's choice is coded as a binary indicator: 1 if the action matches the AEp prediction, 0 if it matches the SPNE/FI prediction. The primary outcome is the aggregate empirical frequency of each prediction across subjects, computed separately per game. In the Modified Reny Game, these two indicators are mutually exclusive by design.

-- Elicited beliefs: subjective probability distribution assigned by Player 1 (and Player 2 in Centipede) over the opponent's available actions.
-- Belief-behavior correspondence: correlation between elicited beliefs and the subject's choice (entry vs. outside option) in BoS-structure games.
-- Within-subject strategic consistency: individual-level classification as SPNE-type, ACBR-type, or Other across the 12 decision blocks.

Beliefs are elicited via an incentivized discrete QSR and recorded as a probability vector over the opponent's actions. Belief-behavior correspondence is measured by testing whether subjects choosing AE actions report p<1 assigned to opponent rationality, using probit/logit regressions, if needed. Individual type classification is constructed under both a strict (100% consistency) and relaxed (such as ≥10/12 decisions) criterion.

-- Study Overview and Subject Pool
This laboratory experiment investigates strategic decision-making and belief formation in sequential environments. We plan to recruit via ORSEE (Greiner, 2015) between 80 and 100 participants. The experiment is programmed and conducted using oTree (Chen et al., 2016).

-- Experimental Tasks
The session consists of 12 rounds of extensive-form games and 7 belief elicitation tasks. The set of games is:
I. One version of the Centipede game with length n=3 and where we elicit strategies with the strategic (cold) method.
II. The two BoS with outside option from Cooper et al. (1993).
III. A Decoy game (with the same structure as the BoS with outside option) with an obvious dominant strategy for both players.
IV. Two modified version of Figure 4 in Reny (1992) (also with the same structure as the BoS with outside option) where SPNE and AE predictions are mutually exclusive.
Subjects are matched in pairs that change for every game (stranger matching) to maintain the one-shot nature of the interactions.
-- Randomization
To control for order effects and learning, we implement a constrained randomization across four pre-defined sequences of play. Participants are randomly assigned to one of these four orders. The sequences are designed following two strict criteria:
I. Participants never play the same game (in different roles) in two consecutive rounds to minimize anchoring and immediate reciprocity effects.
II. All sequences begin with one game designed to test AE vs. SPNE, with the participant in the role of Player 1. This ensures that the primary strategic choice is elicited in a "clean" environment, uncontaminated by learning within the session.

-- Belief Elicitation
Following the decisions, we elicit subjective beliefs using a discretized Quadratic Scoring Rule (QSR). Participants allocate 10 tokens (10% increments) across the opponent’s strategy space. The payoff function is strictly concave, with a maximum reward of €2.00 per task, ensuring incentive compatibility for risk-neutral and risk-averse agents.

-- Payment Protocol
To mitigate wealth effects and hedging, we use a Random Incentive System (RIS). Only 2 out of the 12 game rounds and 2 out of the 7 belief tasks are randomly selected for payment at the end of the session. Additionally, there is a fixed show-up fee of 3 Euros.

Not available

Randomization is performed in the office by a computer using the experimental software oTree.

The order of games and role assignments are randomized at individual level within the software.

No

80-100 individuals

Actions: 960-1200 observations (80-100 participants x 6 games each x 2 player role each). Beliefs: 560-700 observations (80-100 participants × [2 beliefs in Centipede + 1 belief in the other 5 games]

Not applicable (Within-subject design). All participants will complete the entire set of 6 games.

This study serves as empirical support for our Approximate Common Beliefs in Rationality (ACBR) framework. We target a sample of 80 to 100 subjects, which is standard for laboratory experiments focused on testing qualitative theoretical predictions. By using a within-subject design with 12 decisions per participant (playing both roles in 6 games), we generate a high number of observations. This allows us to robustly identify the frequency with which subjects follow our Approximate Equilibrium versus the Subgame Perfect Equilibrium (SPNE). The large number of total observations per participant provides sufficient power to determine if behavior systematically aligns approximate rationalizability.