A Quantum Information Processing Explanation of Disjunction Effects

Abstract

A Quantum Information Processing Explanation of Disjunction Effects Jerome R. Busemeyer (Jerome.Busemeyer@afosr.af.mil) Air Force Office of Scientific Research, 875 N. Randolph Street Arlington, VA 22203 USA Mervin R. Matthew (mermatth@indiana.edu) Zheng Wang (zhenwang@indiana.edu) Department of Psychological and Brain Sciences, 1101 E. 10 th Street Bloomington, IN 47405 USA deciding to cooperate (which is actually the union of the trustworthy and untrustworthy possibilities). In other words, the probability of the disjunction can fall below the probability of a component event, which is a violation of the OR rule within classic probability theory. Has such a violation ever been empirically observed? Abstract A new approach to game theory based on quantum strategies is used to explain some paradoxical phenomena of human choice behavior. Quantum strategies were originally used to explain the fact that humans prefer to cooperate rather than defect in a Prisoner’s Dilemma (PD) game. Here we develop a quantum model for the disjunction effect. This refers to a paradox in which (a) a player prefers to defect when the player knows that an opponent will defect, and (b) the player also prefers to defect when the player knows that an opponent will cooperate, but (c) the player reverses preference and cooperates when the opponent’s action is unknown. New experimental findings on the disjunction effect are reported, and a quantum explanation for the findings is presented. The quantum model is also compared to traditional information processing models. Disjunction Effects Consider a PD game in which there are two players, you versus other, and each player has two actions: cooperate or compete. An example payoff matrix for each player, conditioned on each pair of actions, is shown in Table 1. Table 1: Example PD Game. Other Competes Keywords: quantum model; disjunction effect; Prisoner’s Dilemma Other Cooperates Quantum Information Processing Human reasoning and decision making involves a great deal of vagueness, uncertainty, and conflict. How best to model these characteristics is a fundamental question for information processing theories of cognition. In this paper, we examine a quantum computing approach to this problem (see Nielsen & Chuang, 2000). Consider for example, the decision whether to cooperate or compete with another business on some high tech venture. For example, this other business may have some technical skills that are needed for success. Suppose this decision also depends on whether the other business is trustworthy or untrustworthy. According to a quantum approach, prior to expressing a decision, the decision maker is in a superposition state in which all of the combinations of beliefs about trustworthiness and preferences about cooperation have some potential to be observed. This idea alone is not terribly interesting because any classic information processing theory could also adopt a similar representation. What is interesting is the uniquely quantum idea that possibilities can interfere with each other as if they exist simultaneously in the mental state. In particular, according to quantum theory, the joint probability of believing the other business is trustworthy and deciding to cooperate can be greater than the marginal probability of You Compete You: 10 Other: 10 You: 25 Other: 5 You Cooperate You: 5 Other: 25 You: 20 Other: 20 In the standard version of the game, hereafter referred to as the unknown condition, the players simultaneously select an action without knowledge of the opponent’s selection. Two new manipulations are used to examine the disjunction effect: In one case, you are initially informed that the other player has chosen to compete; and in another case, you are initially informed that the other player has chosen to cooperate. This manipulation is designed to test the ‘sure thing’ principle that lies at the foundation of utility theory (Savage, 1954): If you prefer to compete knowing that your opponent will compete and you prefer to compete knowing that your opponent will cooperate, then you should prefer to compete even when you do not know your opponents choice. Shafir and Tversky (1992) found that players frequently violated the sure thing principle – many players chose to compete knowing that the other player competed, and they also chose to compete knowing that the other player chose to cooperate, but they cooperated when they did not know the choice of the other player. See Croson (1999) and Li and Taplan (2002) for replications and extensions. The disjunction effect also rules out a simple yet important information processing model for this task.