Quasi-magical thinking and the public good

May 12, 2013 by Marcel Montrey

Cooperation is a puzzle because it is not obvious why cooperation, which is good for the group, is so common, despite the fact that defection is often best for the individual. Though we tend to view this issue through the lens of the prisoner’s dilemma, Artem recently pointed me to a paper by Joanna Masel, a mathematical biologist at Stanford, focusing on the public goods game [1]. In this game, each player is given 20 tokens and chooses how many of these they wish to contribute to the common pool. Once players have made their decisions, the pool is multiplied by some factor m (where mn > 1) and the pool is distributed equally back to all players. To optimize the group’s payoff, players should take advantage of the pool’s multiplicative effects by contributing all of their tokens. However, because a player’s share does not depend on the size of their contribution, it is easy to see that this is not the best individual strategy (Nash equilibrium). By contributing nothing to the common pool, a player gets a share of the pool in addition to keeping all of the tokens they initially received. This conflict captures the puzzle of cooperation, which in this case is: Why do human participants routinely contribute about half of their funds, if never contributing is individually optimal?



As Masel points out, various attempts at explaining human cooperation in this context have failed. The proposal that players have not understood the game is contradicted by the finding that cooperation perseveres when subjects play for extended periods, and even resets to high levels when a new round is started [2]. The argument that players cooperate in an attempt to evoke reciprocity also falls flat because subjects who play anonymously, and with no knowledge of their partners’ contributions, not only continue to cooperate, but do so at equal [3, 4] or even higher levels [5]. A final proposal of particular interest to us is the suggestion that players are using a utility function (i.e., considering a payoff matrix) that deviates from objective reality. In other words, they are considering subjective factors such as fairness, the group’s payoff, the rewarding nature of contributing, and so on.

To explain the data and yet stray as little as possible from the assumption of rationality, Masel proposes that human reasoning may be captured by the idea “what if everyone else thought like me?” Specifically, even though players understand there is no causal link between their own behavior and that of others, they may nevertheless recognize that a correlation exists, and this realization may be sufficient motivation to contribute. Famously proposed by Shafir and Tversky [6], this phenomenon is known as quasi-magical thinking and involves acting as if one erroneously believes (without actually believing) that one’s actions affect the behavior of others. This principle may best be captured by the sentiment often expressed by voters, who individually have very little influence on the outcome of any given election, “if I don’t vote, then who will?” In this case, players contribute because they are acting as if they believe that contributing makes others more likely to contribute.

(As an aside, Artem points out that this idea resembles Douglas Hofstadter’s concept of superrationality, a type of decision making where individuals assume that, in a symmetric game, both parties will arrive at the same answer. Because unilateral actions are off limits, this results in cooperation instead of defection, since cooperation is the best mutual strategy. The difference, in this case, is that players do not assume that others will mirror their actions; rather, they are simply sensitive to the fact that their behavior is likely to be correlated to some degree with the behavior of others.)

To test this idea, Masel considers agents using a Bayesian update scheme to estimate how much others contribute and how much these contributions vary. This would ordinarily result in a race to the bottom, with agents converging to the Nash equilibrium (no one contributing any of their tokens). Masel avoids this by having agents treat their own expected contribution as a data point akin to other players’ contributions. This expected contribution is weighted more heavily initially, while an agent’s confidence in its estimate of the average contribution is low, and becomes weighted less heavily relative to external data as time goes on and confidence grows. As a result, agents can increase their estimate of the average contribution simply by expecting to contribute more themselves, particularly when not enough reliable data has been collected to disagree.

In principle, such a bias seems reasonable. It would encourage cooperation, despite cooperation not being individually optimal, and avoids strongly violating the assumption of rationality by explaining the tendency to cooperate as a consequence of what data is used to predict others’ behavior. This broadly agrees with the finding that making choices influences expectations [7] and, conversely, that estimating others’ actions prior to making a choice leads to reduced contributions [8]. In short, leveraging the knowledge that “I am like them” may explain, in rational terms, seemingly irrational cooperation in the public goods game.

References