Risk sensitivity and theory of mind in human coordination
Fig 2
CPT-value difference between certainties and gambles, VCertainty(b) − VGamble(b, α), for different reference points b and probability weighting functions parameterized by Prelec’s parameter α.
The CPT-value is calculated using utility functions u+(x) = x0.85 for gains (x > 0) and u−(x) = −2|x|0.85 for losses (x < 0), and probability weighting functions are of the form w(p|α, δ) = exp{−α(−log(p))δ}, with δ = 0.75. Blue regions indicate larger positive differences, while red regions indicate larger negative differences. The grey solid line represents the decision boundary, where both values are equal. In (a), the agent is choosing between the certainty of gaining 900 and a gamble in which one might gain 1000 with probability 95%, where the optimal choice is ‘Gamble’ since 1000 × 95% = 950 > 900. In (b), the agent is choosing between the certainty of losing 900 and a gamble in which one might lose 1000 with probability 95%, where the optimal choice is Certainty since −1000 × 95% = −950 < −900. In (c), the agent is choosing between the certainty of gaining 55 and a gamble in which one might gain 1000 with probability 5%, where the optimal choice is ‘Certainty’ since 1000 × 5% = 50 < 55. In (d), the agent is choosing between the certainty of losing 55 and a gamble in which one might lose 1000 with probability 5%, where the optimal choice is Gamble since −1000 × 5% = −50 > −55. In all four cases, the reference point and probability perception influence behavior in significant ways. In a) and d), an optimal choice is attained when the agent either has an optimistic view of outcomes (i.e., low reference point b) and overestimates low probabilities (i.e., low Prelec parameter α), or when the agent underestimates high probabilities (i.e., high α) and is pessimistic (i.e., high reference point b). The opposite happens in b) and c). In cases where both certainty and possibility effects are present (i.e., α ≈ 1), behavior becomes extremely non-monotonous as a function of both parameters.