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Supporting Information
Computational theoryofmind model
Player types defined by FehrSchmidt inequity aversion. As stated in the main text, a player type is represented by her degree of inequality aversion. Player i values immediate payoffs using the FehrSchmidt (1999) utility function (eqn 1):
EMBED Equation.3
where EMBED Equation.3 is the money obtained by player i and EMBED Equation.3 is the amount obtained by player j. Two sorts of inequity are important: envy (partner j gets more than subject i ; EMBED Equation.3 in eqn 1a) and guilt (subject i gets more than partner j; EMBED Equation.3 in eqn 1a). The envy and guilt parameters comprise what we consider as the type of a player. Empirically, the majority of investors invest more than half of the endowment and the modal behavior of trustees is to split the sum of money evenly. Hence, the influence of envy on subjects choices was minimal. For simplicity, we assume EMBED Equation.3 QUOTE QUOTE and consider only guilt  the aversion to inequity favorable to the subject as the way to type a player. The utility function becomes:
EMBED Equation.3
Therefore, player is type is fully described by EMBED Equation.3 QUOTE , the guilt parameter. A player only knows her own type but not her opponents type. At any stage of the game, she maintains beliefs about the possible type of her opponent. Moreover, these beliefs are not restricted to first order beliefs about a partners type, but also include beliefs about the partners belief about their type and so on. Such situation is similar to a Partially Observable Markov Decision Process (POMDP) (32) or interactive POMDP (IPOMDP) (33). Here, we extend the POMDP or IPOMDP to a Bayesian game setting.
Utility in the trust game. At round t of the tenround trust game, the investor (with type EMBED Equation.3 ) starting with 20 points decides to send points EMBED Equation.3 to the trustee. The amount is then tripled, and the trustee (with type EMBED Equation.3 ) repays points EMBED Equation.3 back. The resulting payoffs of each player at the end of round QUOTE t are:
Investor: QUOTE EMBED Equation.3
Trustee: EMBED Equation.3
The immediate utility for each player becomes:
Investor: EMBED Equation.3
Trustee: EMBED Equation.3
Q values. Here we write the model for how player QUOTE i forms an estimate of optimal play at each round t by calculating the values EMBED Equation.3 of their possible actions EMBED Equation.3 . The actions are the amounts to invest or to return. The EMBED Equation.3 values are the expected summed utilities over the next two rounds in future. The utility for player i depends on the actions of player j, which in turn depends on the type of player j, and the reasoning that player j does about player i. Player i does not know player js type, but can learn about it from the history of their interactions, which, up to round t is EMBED Equation.3 . Formally, player i maintains beliefs EMBED Equation.3 , in the form of a probability distribution over the type of player j, and computes expected utilities by averaging over these beliefs. Bayes theorem is used to update the beliefs based on evidence.
The EMBED Equation.3 value on round t is a sum of two expectations:
EMBED Equation.3 EMBED Equation.3
The first is the utility of the exchange on that round. This is
EMBED Equation.3
where, for convenience, we write QUOTE EMBED Equation.3 as a function of the possible actions a of player j rather than the money this player earns. The second term in the expectation concerns the value of the future two rounds in the exchange (except in the last round, where this term is 0). This is thus an average over QUOTE QUOTE EMBED Equation.3 values QUOTE EMBED Equation.3 on round t+1, where the new beliefs QUOTE EMBED Equation.3 take account of the action EMBED Equation.3 being considered by player i, and all the possible actions EMBED Equation.3 of player j. Equation (2) is a form of Bellman equation.
All the beliefs are captured with multinomial distributions, and are estimated by simulating the partners play.
Choosing actions. To choose an action, players use a softmax policy based on the QUOTE EMBED Equation.3 values of the stateaction pairs. The Softmax action selection rule is a probabilistic way to go from a set of stateaction values to an action (as opposed to hard max whereby you would always choose the action with greatest value).
The probability of choosing action EMBED Equation.3 given game history EMBED Equation.3 for player QUOTE i is:
EMBED Equation.3
Here QUOTE EMBED Equation.3 is the inverse temperature parameter. It controls how sharp the action selection is. For fixed QUOTE EMBED Equation.3 as EMBED Equation.3 the distribution tends to the uniform distribution. We used two temperature settings: QUOTE EMBED Equation.3 and QUOTE EMBED Equation.3 . For each player, we optimized the temperature according to the best loglikelihood resulted from fitting the model to actual behavior.
Belief representation. As described above, a players type is defined by his guilt coefficient EMBED Equation.3 QUOTE . We discretized ( and assumed EMBED Equation.3 . A belief about type ( described a players uncertainty over the probability of the five ( values, i.e. a probability density ( over weights EMBED Equation.3 , and EMBED Equation.3 QUOTE . The density ( takes the form of the Dirichlet distribution (the conjugate priors for the multinomial distribution):
EMBED Equation.3
where EMBED Equation.3 QUOTE are the hyperparameters, and the normalizing factor EMBED Equation.3 QUOTE is a beta function.
To compute the EMBED Equation.3 values using Eq. 2, a player V assesses the probability of his opponent X s action EMBED Equation.3 given current game history, EMBED Equation.3 QUOTE , which involves computing integrals over the 4simplex belief space:
EMBED Equation.3
We performed the numerical integration using Gaussian quadrature over the belief space.
Depthofthought and belief update. One subtlety that arises is that computing EMBED Equation.3 QUOTE requires QUOTE EMBED Equation.3 , through both its explicit appearance in the Bellman equation and through EMBED Equation.3 , QUOTE is updated beliefs about types, given that QUOTE i chooses EMBED Equation.3 and j chooses EMBED Equation.3 . Consequently, an agent must compute the EMBED Equation.3 QUOTE values of their opponent, leading to an infinite regress. To avoid this, we assume that a player reasons at a hierarchical level (as in cognitive hierarchy theory32), and assume their opponent reasons at one level lower. Players can have three cognitive levels, or depthofthought. Level 0 player QUOTE i simply computes the immediate utility value of player QUOTE j, EMBED Equation.3 , and uses it to compute EMBED Equation.3 QUOTE through the softmax function, and updates their beliefs. A level 1 player assumes that her partner QUOTE j is level 0, and simulates QUOTE js play by computing EMBED Equation.3 . Similarly, a level 2 player assumes the partner is level 1, and computes her partners EMBED Equation.3 QUOTE values to get EMBED Equation.3 QUOTE . Beliefs are updated as follows:
A level 0 player QUOTE i does not simulate the opponents play. It computes the likelihood of observing opponents action EMBED Equation.3 using the immediate utility for five possible guilt coefficients EMBED Equation.3 :
EMBED Equation.3
A level 1 or 2 player QUOTE i ( EMBED Equation.3 = 1 or 2, depthofthought) regards the opponent as a level QUOTE EMBED Equation.3 player and simulates her play. It computes the likelihood of observing opponents action QUOTE EMBED Equation.3 using the EMBED Equation.3 value:
EMBED Equation.3
The belief update follows Bayes rule by updating the hyperparameters ( of the Dirichlet distribution. We set the prior belief as: EMBED Equation.3 QUOTE . The posterior belief is given by a Dirichlet distribution with hyperparameters given by:
EMBED Equation.3
Behavioral classification. We then applied the statistical inverse of the generative model above and described in the main text to classify individual players according to their behavior in the sequential trust game.
Since the game is Markovian, we can calculate the likelihood of player i taking the action sequence EMBED Equation.3 given her type QUOTE EMBED Equation.3 , prior beliefs QUOTE EMBED Equation.3 and depthofthought EMBED Equation.3 as:
EMBED Equation.3
where EMBED Equation.3 QUOTE is the probability of initial action EMBED Equation.3 given by the softmax distribution, prior beliefs EMBED Equation.3 QUOTE and depthofthought EMBED Equation.3 , and EMBED Equation.3 is the probability of taking action EMBED Equation.3 after updating beliefs EMBED Equation.3 QUOTE from previous beliefs EMBED Equation.3 QUOTE upon observing the history of moves EMBED Equation.3 QUOTE .
Finally, we classify the players for their type QUOTE EMBED Equation.3 and depthofthought EMBED Equation.3 by finding values of that maximize EMBED Equation.3 .
Reinforcement learning model
For comparison with a perhaps conceptually simpler model, we constructed a reinforcement learning model. As in the computational theoryofmind model, we used the FehrSchmidt utility function and only considered the guilt coefficient. So a players immediate utility or reward was given by
EMBED Equation.3
where EMBED Equation.3 is the players immediate payout, and EMBED Equation.3 is the other players immediate payout.
We considered five possible actions for investors and trustees. Investors actions EMBED Equation.3 QUOTE , and Trustees actions EMBED Equation.3 QUOTE . In this reinforcement learning model, players learned a value associate with each action. For improved generalization, since the number of rounds was limited, players learned a linear function for the actionvalues:
EMBED Equation.3
At round t, player EMBED Equation.3 QUOTE computed a prediction error EMBED Equation.3 QUOTE after observing the opponent EMBED Equation.3 QUOTE s action:
EMBED Equation.3 QUOTE .
After calculating the prediction error, the value was then updated according to the temporal difference rule: EMBED Equation.3 QUOTE , where ( QUOTE was the learning rate. The best parameters EMBED Equation.3 QUOTE for the linear fit were then recalculated using the new value of the chosen action and the old values of the unchosen actions to get EMBED Equation.3 QUOTE . The values for all the actions were then updated with the new parameters EMBED Equation.3 QUOTE .
To fit this model to the actual behavior, we computed the probability of selecting action EMBED Equation.3 QUOTE using a softmax likelihood, given the actionvalue EMBED Equation.3 QUOTE :
EMBED Equation.3
The negative loglikelihood of the game history D given the parameters EMBED Equation.3 QUOTE was given by:
EMBED Equation.3
The values of k ranged between 0 and 30, and the values of b ranged between 0 and 20. The learning rate EMBED Equation.3 QUOTE . For each subject pair, we calculated the negative loglikelihood, and chose the best fit values of EMBED Equation.3 QUOTE and EMBED Equation.3 QUOTE . We then took the averaged negative loglikelihoods over all subject pairs within a group (we had four groups, Impersonal, Personal, BPD and BPD controls).
We report the best fitting parameters k and b and the averaged negative loglikelihood for different learning rates in Table S1 and Table S2, respectively. We found that the parameters maximizing the loglikelihood were EMBED Equation.3 QUOTE , and EMBED Equation.3 QUOTE . Note that if all the five actions were chosen with equal probability, the negative loglikelihood would take the value EMBED Equation.3 QUOTE . Thus the model degenerated to the case where the values were uniform, no learning occurred, and all actions were selected equally. Table S2 also includes the negative loglikelihood for the computational theoryofmind model. Comparison demonstrates that the computational theoryofmind model provides a better fit.
References
Kaelbling LP, Littman ML, Cassandra AR (1998) Planning and acting in partially observable stochastic domains. Artificial Intelligence 101:99134.
Gmytrasiewicz PJ, Doshi PA (2005) Framework for Sequential Planning in Multiagent Settings. JAIR 24:4979.
Xiang et al.
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