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Source: http://www.doksinet 4 Effects of Individual Decision Theory Assumptions on Predictions of Cooperation in Social Dilemmas Van Assen, M.ALM (1998) Effects of individual decision theory assumptions on predictions of cooperation in social dilemmas. Journal of Mathematical Sociology, 23, 143-153. Source: http://www.doksinet 68 Chapter 4 Abstract Raub and Snijders (1997) show that, under the assumption of S-shaped utility, conditions for cooperation in social dilemmas are more restrictive if outcomes represent gains than if outcomes represent losses. They neglected two interesting issues in their paper: conditions for cooperation in social dilemmas with both losses and gains as outcomes, and the effect of probability weighing on these conditions. In this comment it is shown that, under assumptions of Prospect Theory, conditions for cooperation are best if dilemmas include both positive and negative outcomes, and that these conditions improve with increasing loss aversion.

Furthermore, it is shown that probability weighing can effect conditions to cooperate as well. Key Words Social dilemma, prospect theory, loss aversion, risk preferences, decision weights Acknowledgements I thank Alexander Gatig for providing me with an earlier draft of Raub and Snijders (1997), and I thank Werner Raub, Chris Snijders, and Peter Wakker for making comments on an earlier draft of this paper. Finally, I thank Sasa Bistrovic for pointing out an inaccuracy in the description of the proof in the appendix. Source: http://www.doksinet Effects of Individual Decision Theory Assumptions on Predictions of Cooperation 4.1 69 Introduction In their first article on Prospect Theory, Kahneman and Tversky (1979) reported substantial empirical evidence for three regularities in individual decision making: S-shaped utility, loss aversion, and probability weighing. S-shaped utility refers to convex and concave utility functions for, respectively, losses and gains. Loss aversion

refers to the relative steepness of the utility function for losses in comparison to gains. Probability weighing refers to the observation that individuals overweigh small and underweigh large probabilities It is argued in this note that S-shaped utility, loss aversion, and probability weighing have important implications for cooperation in social dilemmas. Raub and Snijders (1997) compared conditions for cooperation in strictly positive and strictly negative repeated PDs. They derived that, under the assumption of S-shaped utility, conditions for cooperation are more restrictive if outcomes represent losses than if outcomes represent gains, i.e, that a gain among smaller gains is a more powerful motivator to cooperate than a loss motivates actors to cooperate to obtain a smaller loss. Raub and Snijders however neglected two interesting issues in their paper: conditions for cooperation in (mixed) PDs with both losses and gains, and the effect of probability weighing on these

conditions. Both issues are addressed in this paper More specifically, it is shown in Section 4.2 that: (1) conditions for cooperation in mixed PDs improve with increasing loss aversion; (2) in well-defined circumstances, assuming S-shaped utility, conditions for cooperation in mixed PDs are less restrictive than for strictly positive (and strictly negative) PDs. Hence in mixed social dilemmas a loss can be a more powerful motivator to cooperate than a gain and loss aversion increases the motivating power of a loss in these dilemmas. In Section 4.3 it is demonstrated how probability weighing can effect the willingness to cooperate: (3) In strictly positive and strictly negative repeated PDs, assuming probability weighing, subjects are less (more) inclined to cooperate if the PDs continuation probability is large (small) than can be expected from the value of the probability alone. The latter result can explain the absence of high cooperation rates in Raub and Snijders experiment.

It is also demonstrated that Raub and Snijders use of a utility assessment procedure not taking probability weighing into account can explain why only a minority of subjects seemed to choose in accordance with S-shaped utility. 4.2 The effects of loss aversion on cooperation in social dilemmas Following Raub and Snijders an indefinitely repeated standard 2-person PD1 (Γ:∆) is considered as an example of a social dilemma. Using their notation, the outcomes of the PD are denoted by T for unilateral defection, R for mutual cooperation, P for mutual defection, and S for unilateral cooperation. By definition of the PD, S < P < R < T The set (Γ:∆) contains all PDs with outcomes T+∆, R+∆, P+∆ > S, where ∆ and S are realvalued constants with S< P+∆. The PD (Γ:∆) is indefinitely repeated in rounds 1,2,,t with a constant probability of ω 1 The results of their derivations and the derivations in this text can be generalized to n-person dilemma games of the

Schelling-type (1978). Source: http://www.doksinet 70 Chapter 4 to play the game at round t+1 given that round t has been played, reflecting exponential discounting of game pay-offs. It is assumed that both actors have complete information and common knowledge with respect to the structure of the repeated PD, and that they are informed on the behavior of the other actor in all previous rounds 1,2,.,t-1 Furthermore, they are assumed to behave in accordance with the same utility function u. A well-known result in game-theory (e.g, Friedman, 1986, pp 88-89) is that a (subgame perfect) equilibrium exists such that actors choose to cooperate in all rounds if and only if ω ≥ ω u ( Γ : ∆ )= u(T + ∆ ) - u(R + ∆ ) u(T + ∆ ) - u(P + ∆ ) (1) In the sequel, the behavior of ωu(Γ:∆) is studied as a function of ∆ without Raub and Snijders restriction to strictly positive or strictly negative PDs (outcomes T+∆, R+∆, and P+∆ having identical sign). Without loss of

generality, outcome P is fixed at value 0 S-shaped utility is assumed together with loss aversion. Loss aversion is modelled by assuming that u(-x) = λu(x) for all x ≥ 0, with λ > 1 The higher the value of λ the more loss aversion To acknowledge that ωu is dependent on the value of the loss aversion parameter, ωu is denoted by ωu(Γ:∆,λ). To examine the behavior of ωu(Γ:∆,λ) four domains of ∆ are distinguished: ∆1 ≥ 0, 0 > ∆2 ≥ -R, -R > ∆3 ≥ -T, and -T > ∆4. Raub and Snijders remark that ωu(Γ:∆1,λ) and ωu(Γ:∆4,λ) are independent from the value of λ. Our first result, however, states that conditions for cooperation in mixed PDs improve with increasing loss aversion. ω u ( Γ : ∆ 2 ,λ ) = u(T + ∆2 ) - u(R + ∆ 2 ) u(T + ∆ 2 ) + λu(- ∆ 2 ) (2) ω u ( Γ : ∆3 , λ ) = u(T + ∆ 3 ) + λu(-R - ∆ 3 ) u(T + ∆ 3 ) + λu(- ∆ 3 ) (3) The result is easily proved: both ωu(Γ:∆2,λ) and ωu(Γ:∆3,λ) are strictly

decreasing in λ, which demonstrates that loss aversion indeed promotes cooperative behavior in mixed PDs. More interesting is a comparison of ωu(Γ:∆2,λ) and ωu(Γ:∆3,λ) to ωu(Γ:∆1,λ) to see whether conditions for cooperative actions are less restrictive in a mixed PD than in a strictly positive PD. Although some results can be derived without assuming a parametric form of the utility function, more can be said about the behavior of ωu(Γ:∆,λ) as a function of its parameters if additional assumptions are made on the utility function. Tversky and Kahneman (1992) found empirical support for a utility function from the power family. Following Tversky and Kahneman (1992) the power family is assumed, but with the restriction that the power parameters for losses and gains are identical: 2 u(x) = xα (x ≥ 0), and u(x) = -λ(-x)α (x < 0). (4) Parameter α is in the interval (0,1) because S-shaped utility is assumed. Since ωu is dependent on α as well, it is a

function of three parameters and is from now on denoted by ωu(Γ:∆,λ,α). 2 In their 1992 experiment Tversky and Kahneman found identical estimates for the power parameter for losses and gains. Source: http://www.doksinet Effects of Individual Decision Theory Assumptions on Predictions of Cooperation Figure 4.1: 71 ωu(Γ:∆,λ,0.88) as a function of ∆ and λ, for ωL = 075 (T = 16, R =12), and ωL = 0.25 (T = 16, R =4) The second result of this section concerns the behavior of ωu(Γ:∆2,λ,α) in relation to ωu(Γ:∆1,λ,α) and is stated in the following theorem. The proof of the theorem and an extension are presented in Appendix 4.1 Theorem ωu(Γ:∆2,λ,α) < ωu(Γ:∆1,λ,α) for all ∆2 in the interval (- RT ,0), and for all ∆1, 0 < α < 1 and λ = 1. R +T Because ωu(Γ:∆2,λ,α) and ωu(Γ:∆3,λ,α) decrease in λ (see (2) and (3)), both the left border of the interval and the numbers [ωu(Γ:∆2,λ,α) - ωu(Γ:∆1,λ,α)] and

[ωu(Γ:∆3,λ,α) - ωu(Γ:∆1,λ,α)] decrease if λ is increased. Together with the results on the effect of loss aversion the Theorem RT ,0) if both do not cooperate, implies that if both actors receive a loss in the interval ( R +T they are more prone to cooperate than in strictly positive PDs (Γ:∆1) and that they are even more so for increasing loss aversion. Thus, in social dilemmas with both negative and positive outcomes a loss can be a more powerful motivator to cooperate than a gain and loss aversion increases the motivating power of a loss in these dilemmas. Unfortunately, no general results of ωu(Γ:∆3,λ,α) in comparison to ωu(Γ:∆1,λ,α) could be derived independently from the values of α, λ, and ∆3. For some values of these parameters ωu(Γ:∆3,λ,α) is less and for others greater than ωu(Γ:∆1,λ,α) for all ∆1. To clarify the results derived in this section, the behavior of ωu(Γ:∆,λ,α) as a function of its parameters is illustrated by

examples, all depicted in Figure 4.1 The two horizontal curves represent distinct values for ωL = ωu(Γ:∆,1,1); ωL = 0.25 (T = 16, R = 12), and ωL = 0.75 (T = 16, R = 4) Three curves for each of the two values are shown in the figure, all with Source: http://www.doksinet 72 Chapter 4 α = 0.88, the value found in the Tversky and Kahneman (1992) experiment for both losses and gains. For each ωL, the upper curve, middle, and lower curve represent the behavior of ωu(Γ:∆,λ,0.88) for respectively λ = 1 (no loss aversion), λ = 225, and λ = 5 The value λ = 2.25 was found by Tversky and Kahneman (1992) as well If the Theorem is applied to the examples it is found that ωu(Γ:∆2,λ,α) < ωu(Γ:∆1,λ,α) 64 192 if λ = 1 and ∆2 ε ( ,0) for ωL = 0.75 and ∆2 ε ( ,0) for ωL = 025 If it is assumed that 20 28 the values for α and λ found by Tversky and Kahneman are correct, then the intervals have more negative boundaries; by calculation it is obtained that the

intervals are (-13.519,0) and (-9.150,0) for respectively ωL = 025 and ωL = 075 Note that these intervals contain the outcome -R, which means that conditions for cooperation in PDs with three negative outcomes (S, P, and R) can be less restrictive than in strictly positive dilemmas. The results of Raub and Snijders and the results derived in this section can be summarized in other words as follows: People are more inclined to cooperate when they are more risk averse. Since it is assumed that people are risk averse for gains and risk seeking for losses, more cooperation is expected for strictly positive than for strictly negative PDs. Loss aversion causes people to be more risk averse for some mixed PDs, and therefore people are more inclined to cooperate in the latter PDs. 4.3 Probability weighing In investigating the effects of probability weighing on both the results derived in Section 4.2 and the results of Raub and Snijders, it is assumed that decision makers weigh

probabilities as formulated in Cumulative Prospect Theory (CPT: Tversky and Kahneman, 1992, pp.300301) In CPT different weight functions π for probabilities belonging to losses and gains are assumed. A property of the functions π is that for strictly positive or strictly negative twooutcome gambles the decision weight functions assign the decision weight π(p) to the probability p belonging to the largest absolute outcome, and 1-π(p) to the probability assigned to the smallest absolute outcome. Note that the property implies that both decision weights add to one. It is assumed, in agreement with empirical results from numerous studies (eg, Kahneman and Tversky, 1979; Tversky and Kahneman, 1992; Camerer, 1995; Camerer and Ho, 1994) that subjects overweigh rather small probabilities (say, smaller than 0.30) and underweigh moderate to large probabilities (say, larger than 0.50) belonging to the largest absolute outcome. Subjects in Raub and Snijders experiment participated in strictly

positive and strictly negative dilemmas which had a probability of 0.75 of continuing If the PD is continued subjects gain (loose) an additional amount of money since all outcomes of a single trial of the PD are positive (negative). Therefore, it is assumed that the decision weight for ω is equal to π(ω), the decision weight for the largest absolute outcome. Assuming that subjects have the same decision weight function π, it can be derived that cooperation is feasible, if π ( ω ) ≥ ωu( Γ )= u(T) - u(R) . u(T) - u(P) (5) The formula above is (1) with ∆ = 0 and ω replaced by π(ω). From the formula above our third result can be derived: Because of overweighing (underweighing) of a small (large) probability ω, subjects are more (less) inclined to cooperate in strictly positive and strictly Source: http://www.doksinet Effects of Individual Decision Theory Assumptions on Predictions of Cooperation 73 negative PDs than can be expected from probability ω alone. In the

next paragraph this result is used to explain one of Raub and Snijders findings that cannot be explained without probability weighing. Raub and Snijders choose the outcomes T, R and P in their experiment such T -R . Raub and Snijders (Table 1) found cooperation rates smaller than 050 for that ω = T -P three out of four groups with risk averse response patterns for losses or gains. Their results can be explained if π(0.75) < ωu(Γ) < 075 Assuming the value function of Tversky and Kahneman (1992) with α = 0.88 and the outcomes used in Raub and Snijders (T = 16, R = 7, P = 4), it can be calculated that ωu(Γ) = 0.733 Hence a small underweighing of ω is sufficient to explain the absence of high cooperation rates in the experiment of Raub and Snijders. In the analysis above it was assumed that the utility function was correctly measured. However, some critical comments can be made on the measurement of utility function characteristics by Raub and Snijders. They assessed whether

the functions for gains (losses) were concave by asking the subjects to choose between the two options; (a) receiving R+ (R-), (b) receiving P+ (P-) with probability ω and T+ (T-) with probability 1-ω. They argue that subjects with concave utility functions will choose the certain option. Raub and Snijders (text under Table 1) however found ". that - contrary to Kahneman and Tverskys conjecture only 23% of all participants chose in accordance with S-shaped utility" Their argument and statement is not correct if probability weighing is assumed. According to CPT the probability 1-ω (0.25) assigned to outcomes T+ and T- is overweighed3 which implies that subjects can prefer the risky option above the certain option, i.e, that they are risk seeking, even if they have a concave utility function. The latter can both explain the unexpected high number of risk seeking response patterns for both losses and gains (44%, Table 1) and the rather small percentage (23%) of expected

response patterns. Thus the results of Raub and Snijders in Table 1 need not be in conflict with the assumption of S-shaped utility and in principal but can be explained by probability weighing as in CPT. If the method of Raub and Snijders is not valid to measure characteristics of the utility function, then what method is? All traditional utility assessment methods (see Farquhar, 1984, for a general overview) assume that subjects do not transform probabilities. Only recently, Wakker and Deneffe (1996) developed a method that can measure utility functions independently from probability transformations. Applications of the method and a discussion of its pros and cons in comparison with traditional utility assessment methods can be found in Wakker and Deneffe (1996) and van Assen (1996). 4.4 Discussion Results from the field of individual decision making are seldomly used in deriving predictions on both subjects and collective behavior in social dilemma-like situations (Raub and

Snijders, 1997). That is surprising, since this and Raub and Snijders paper clearly show that results from research in the field of individual decision making, like S-shaped utility, loss aversion, and probability weighing, have substantial implications for predictions of cooperation in social dilemmas. 3 The probabilities assigned to T and T- are overweighed since, respectively, both π(0.25) > 025 and 1-π(075) + > 0.25 Source: http://www.doksinet 74 Chapter 4 The results derived in both papers suggest applications as well. Actors evaluate outcomes in terms of gains and losses relative to a (subjective) reference point rather than in terms of final outcomes. The reference point, however, is dependent on the description of the choice situation (e.g, Kahneman and Tversky, 1979) Thus, if a group of actors is better off when its members cooperate, their leader(s) can enhance conditions for cooperation by framing the situation as a situation in which defection leads to a loss,

and cooperation to a larger gain or a smaller loss. Since the application of individual decision theory to repeated PDs yielded new insights in conditions for cooperative behavior, it can be fruitful to apply individual decision theory to other social dilemmas as well. The predictions resulting from these analyses can then be tested in empirical studies like in Raub and Snijders (1997). Source: http://www.doksinet Effects of Individual Decision Theory Assumptions on Predictions of Cooperation 75 Appendix 4.1: Proof of Theorem The proof of the Theorem consists of several steps: (i) calculate the minimum of ωu(Γ:∆1,λ,α), (ii) proof that ωu(Γ:∆2,λ,α) = min[ωu(Γ:∆1,λ,α)] if ∆2 = - (iii) proof that ωu(Γ:∆2,λ,α) < min[ωu(Γ:∆1,λ,α)] for all ∆2 ε ( - RT for λ = 1 and all α ε (0,1), R +T RT ,0) for λ = 1 and all α R +T ε (0,1). (i) Since ωu(Γ:∆1,λ,α) is increasing in ∆1 (see Theorem 3 of Raub and Snijders) the minimum of

ωu(Γ:∆1,λ,α) is obtained for ∆1 = 0. Hence the minimum is equal to T α - Rα Tα (ii) (6) Modify (2) by substituting λ = 1 and using (4). Then equate the modified equation (2) to (6) to obtain after some manipulations: α α α α Rα (- ∆ 2 ) - T α (R + ∆ 2 ) - T α (- ∆ 2 ) + Rα (T + ∆ 2 ) = 0 Substituting ∆2 = (iii) (7) RT in (7) results in the equality 0 = 0. R +T It is sufficient to show that the second derivative of (the left hand side of) (7) with respect to ∆2, which is the numerator of the difference ωu(Γ:∆2,λ,α) - min[ωu(ΓRT :∆1,λ,α)], is positive for all ∆2 ε ( ,0). Differentiating (7) twice, dividing by (1R +T α) and rearranging terms yields α - α α α R T + - R 2 -α + T 2 -α 2 -α 2 -α (T + ∆ ) (R + ∆ ) (- ∆ ) (- ∆ ) (8) The value of (8) is clearly larger than zero for all ∆2 and thus also for all ∆2 ε RT (,0), because the positive terms are larger than the negative terms. R +T A result more general

than the Theorem can be derived by extending the set of PDs * (Γ:∆1) to all strictly positive PDs (Γ :∆1) with the same ωL, i.e, with outcomes T = aT, R = aR, and a > 0. Because the minimum of ωu(Γ*:∆1,λ,α) (obtained in ∆1 = 0) is equal to the minimum of ωu(Γ:∆1,λ,α), the Theorem applies to set (Γ*:∆1) as well