Penalties and Rewards As Inducements To Cooperate Λ Cheng-Zhong Qin y September 3, 2002 Abstract This paper considers two mechanisms for promoting coo

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1 Penalties and Rewards As Inducements To Cooperate Λ Cheng-Zhong Qin y September 3, 2002 Abstract This paper considers two mechanisms for promoting cooperation in prisoner's dilemma, one penalizes defection and the other rewards cooperation. The paper characterizes defection-penalties and cooperation-rewards inducing the players to cooperate. Payoffs for prisoner's dilemma can be partitioned according to whether both mechanisms promote cooperation, one but not both mechanisms promotes cooperation, and neither of the two mechanisms promotes cooperation. KEYWORDS: Mutual cooperation, prisoner's dilemma, penalty, reward, subgameperfect equilibrium. (JEL C72, K12) 1. Introduction A fundamental characteristic of prisoner's dilemma is that eachoftwoplayers could capture substantial gains through cooperation but is tempted by even greater gains should the player act selfishly, while the other player acts cooperatively. For either player, the worst case is to cooperate while the other defects. The result is that both players defect, even though joint defection leaves each player with less than the player could have obtained through mutual cooperation. Λ An earlier version of this paper was titled Credible Commitments: Ex Ante Deterrence and Ex Post Compensation". I wish to thank Chong-En Bai, Ted Bergstrom, Gary Charness, Francis Cheung, Stephen Chiu, Ted Frech, Rod Garratt, Matthew Jackson, Tom Palfrey, Charles Stuart, Guofu Tan, Zhigang Tao, Joel Watson, Lin Zhou, and participants at the Midwest Theory Conference at University of Minnesota, October 13-15, 2000, the South West Economic Theory Conference at Caltech, March 2-4, 2001, and seminar participants at UBC, UCR, and the University of Hong Kong for helpful comments and suggestions. y Department of Economics, University of California, Santa Barbara, CA

2 This paper considers a penalty mechanism and a reward mechanism for promoting cooperation in prisoner's dilemma. The penalty mechanism allows each player to preselect apayment which the player will make to the other player if he or she defects while the other player cooperates. We call such conditional payments defection-penalties. The penalty mechanism transforms a prisoner's dilemma game into a two-stage game: The players select defection-penalties in stage 1 and in stage 2, knowing each other's preselected defectionpenalties, the players decide what actions to take in the prisoner's dilemma game. We say that a pair of defection-penalties, one for each player, induces the players to cooperate if it is a subgame-perfect equilibrium outcome for the players to select defectionpenalties as in the pair in stage 1 and to cooperate in stage 2 conditional on those defectionpenalties. We establish necessary and sufficient conditions for a defection-penalty pair to induce the players to cooperate. These conditions require that the defection-penalty from player 1 be, on the one hand, large enough to deter player 1 herself from defecting and, on the other hand, not so large that player 2 would rather have player 1 defect than have both of them cooperate. The defection-penalty from player 2 is subject to similar upper and lower bounds. The reward mechanism allows each player to preselect a payment which the player will make to the other player if he or she cooperates. We call such conditional payments cooperation-rewards. 1 As with the penalty mechanism, the reward mechanism transforms a prisoner's dilemma game into a two-stage game: The players select cooperation-rewards in stage 1 and in stage 2, knowing each other's preselected cooperation-rewards, the players decide what actions to take in the prisoner's dilemma game. We say that a pair of cooperation-rewards, one for each player, induces the players to cooperate if it is a subgame-perfect equilibrium outcome for the players to select cooperation-rewards as in the pair in stage 1 and to cooperate in stage 2 conditional on those cooperation-rewards. We esatablish necessary and sufficient conditions for a cooperation-reward pair to induce the players to cooperate. These conditions require that the cooperation-reward from player 1 be, on the one hand, large enough to induce player 2 to cooperate and, on the other hand, not so large as to cause one of the following two cases to occur: In the first case, the cooperation-reward is so large that player 1 would rather 1 This reward mechanism is adapted from the simplified compensation mechanism of Varian (1994, pp. 1291). Varian shows, among other things, that the simplified compensation mechanism implements the efficient outcome of the prisoner's dilemma with certain specifications of the payoffs in subgame-perfect equilibrium. Ziss (1997) shows that the efficient outcome is not among the set of possible subgameperfect equilibrium outcomes for certain other specifications of the payoffs. In a recent experimental study of whether subjects actually manage to achieve mutual cooperation with the help of the simplified compensation mechanism, Andreoni and Varian (1999) finds support for the mechanism. 2

3 have player 2 defect than have him cooperate. In the second case, the cooperation-reward is so large relative to player 2's cooperation-reward to player 1 that she prefers mutual defection to mutual cooperation. The cooperation-reward from player 2 is subject to similar requirements. Our characterization of defection-penalties and cooperation-rewards inducing the players to cooperate implies a partition of the payoffs for prisoner's dilemma according to whether (i) both the penalty mechanism and the reward mechanism promote cooperation, (ii) the penalty mechanism promotes cooperation but the reward mechanism does not, (iii) the reward mechanism promotes cooperation but the penalty mechanism does not, and (iv) neither of the two mechanisms promotes cooperation. 2 The rest of the paper is organized as follows. Section 2 introduces prisoner's dilemma. Section 3 characterizes defection-penalties and cooperation-rewards inducing the players to cooperate. Section 4 compares defection-penalties with cooperation-rewards as inducements to cooperate and section 5 summarizes the paper. 2. Prisoner's Dilemma A generic prisoner's dilemma game is represented by the payoff matrix in the following figure, where C and D stand respectively for the actions to cooperate and to defect: [Insert FIGURE 1 here] When two parties have promised something to each other without enforcement, each party has incentives to take the benefit of the other party's performance while refusing to perform his or her own. The result is that neither will perform, when both prefer mutual performance to mutual breach. The obvious fear is that performance by one party and breach by the other leaves the performing party worse off than breaching as well. That is, the two parties find themselves faced with a prisoner's dilemma game. Consider for example an economic transaction in which you agree to provide money in exchange for a car. Suppose, however, that the delivery of each takes time. The present owner will not know that he is getting his money until after he has shipped the car; you will not know that you are getting the car until the money is sent. But you need the car more than the money, and he needs the money more than the car. Hence each ofyou gains when you both stick to the agreement (the action to cooperate). However, you gain more if you cheat on the agreement (your action to defect), while the present owner sticks to the agreement. Likewise, the present owner gains more if he cheats on the agreement (his 2 Williamson (1983, pp ) discusses the merit of crafting ex ante incentive structures for prisoner's dilemma. 3

4 action to defect), while you stick to the agreement. You each therefore find yourself in a prisoner's dilemma game with the other (Campbell 1985, pp. 9). The following payoff changes of the players are used in our characterization of defectionpenalties and cooperation-rewards inducing the players to cooperate. 3 First, player 1 can guarantee herself at least payoff P 1 by defecting. In cooperating and in trusting player 2 to cooperate, she may get P 1 S 1 units less than she could guarantee herself by defecting. This payoff decrease is player 1's risk" to cooperate. Second, if player 2 is going to cooperate, player 1 can gain T 1 R 1 units more by defecting. This payoff increase is player 1's temptation" to defect. Third, when both players cooperate, player 1 gains R 1 P 1 units more than what she can guarantee herself. This payoff increase is player 1's gain" for mutual cooperation. Player 2's risk to cooperate, temptation to defect, and gain for mutual cooperation are specified similarly. 3. Promoting Cooperation via the Penalty and the Reward Mechanisms In this section we characterize defection-penalties and cooperation-rewards inducing the players to cooperate in prisoner's dilemma games with transferable payoffs Promoting Cooperation Via the Penalty Mechanism Under the penaltymechanism a player gets penalized byhaving to pay his or her preselected defection-penalty to the other player if only he or she defects. We quantify defectionpenalties from a player by the payoff reductions that they impose upon the player. We assume that defection-penalties from player 1 increase player 2's payoff by the amounts equal to the payoff reductions these defection-penalties impose upon player 1, and vice versa. Denote by H 1 and H 2 the sets of defection-penalties that players 1 and 2 can respectively preselect. We first assume that no player pays any defection-penalty intheevent ofmutual defection. In the simple example of the one-shot sale of the car, mutual defection means that you do not send the money and the present owner does not deliver the car. Since neither of you incurred any expense in reliance on the agreement, you are both back tothe same position you were before the agreement was made. This is the outcome that would be achieved if both of you refuse to deal with each other. Thus, for such situations, the assumption that no player pays any defection-penalty in the event ofmutual defection may 3 These payoff changes were used before in the literature on experimental study of the propensity to cooperate in symmetric prisoner's dilemma (see Bonacich 1970). 4

5 be justified by considering mutual defection as refusal to transact in the expectation of being defected. With the penalty mechanism a strategy for player 1 must specify a defection-penalty that she will preselect and in addition, the strategy must also specify how her action in prisoner's dilemma reacts to the various defection-penalties that they each may preselect. We call such reactions player 1's action plans. Equivalently, an action plan for player 1 is a mapping Φ 1 that maps each defection-penalty pair H 2 H 1 H 2 into a probability distribution Φ(H)over the action set fc; Dg. Let Φ 1 (C; H) denote the probability assigned to action C by action plan Φ 1 conditional on defection-penalty pair H 2 H 1 H 2. By analogy, a strategy for player 2 must specify a defection-penalty that he will preselect and an action plan specifying how his action in prisoner's dilemma reacts to the arious defection-penalties that they each may preselect. A defection-penalty pair H changes the prisoner's dilemma game in Figure 1 into game p (H) infigure 2 below, which is the subgame following defection-penalty pair H. 4 [Insert FIGURE 2 here] We consider the possibility for inducing the players to cooperate via defection-penalties in subgame-perfect equilibrium. In general, a subgame-perfect equilibrium is a strategy profile that remains a Nash equilibrium for any subgame including the game itself with the strategy profile restricted to the subgame. In our case, a subgame-perfect equilibrium is a strategy profile ((H Λ 1; Φ Λ 1); (H Λ 2; Φ Λ 2)) such that the strategy profile is a Nash equilibrium for the two-stage game and in addition, for any defection-penalty pair H, the (randomized) action pair (Φ Λ 1(H); Φ Λ 2(H)) is a Nash equilibrium for the subgame p (H). Subgame perfection eliminates those solutions that rest on the players' action plans that are not credible. DEFINITION 1: A defection-penalty pair H Λ =(H Λ 1;H Λ 2) induces the players to cooperate if there are action plans Φ Λ 1 and ΦΛ 2 such that ΦΛ 1(C; H Λ ) = Φ Λ 2(C; H Λ ) = 1 and such that the strategy profile ((H Λ 1; Φ Λ 1); (H Λ 2; Φ Λ 2)) is a subgame-perfect equilibrium for the two-stage game. That is, a defection-penalty pair H Λ = (H Λ 1;H Λ 2) induces the players to cooperate if, as a subgame-perfect equilibrium, player 1 selects H Λ 1 and player 2 selects H Λ 2 in stage 1 and they both cooperate in stage 2 conditional on these defection-penalties. We say that the penalty mechanism promotes cooperation if there exists a defection-penalty pair inducing 4 When the defection-penalty pair H is clearly understood, we will omit the phrase following defectionpenalty pair H". 5

6 the players to cooperate. We assume that the players can select any nonnegative defectionpenalties. That is, we assume H 1 =[0; 1) andh 2 =[0; 1). The inducibility of cooperation in prisoner's dilemma via defection-penalties is not warranted. Indeed, cooperation in the following prisoner's dilemma game is not inducible. A. An Example Consider a prisoner's dilemma game with payoffs as in the following figure: [Insert FIGURE 3 here] Notice that mutual cooperation is efficient in this game, in the sense that it yields the highest total payoff. From Figure 2, a defection-penalty pair H leads to the following subgame p (H): [Insert FIGURE 4 here] Fix player 2's defection-penalty H 2 with 6 <H 2 < 10. Then from Figure 4, action C is strictly dominant forplayer 1 when H 1 > 5; and action D is strictly dominant for player 2 when H 1» 5. This implies that, for any H 1 2H 1 and for any H 2 2H 2 with 6 <H 2 < 10, action pair (C; D) is the only Nash equilibrium for p (H). Thus, given that player 1 plays a subgame-perfect equilibrium strategy, player 2 receives payoff 30 H 2 > 20 by preselecting a defection-penalty H 2 strictly between 6 and 10 and by defecting in subgames p (H 1 ;H 2 ), H 1 2 H 1. This shows that no defection-penalty pair can induce the players to cooperate, because player 2's payoff would be exactly 20 were they induced to cooperate. The reason why the penalty mechanism fails to promote cooperation in this example is because player 2 can preselect a defection-penalty that covers player 1's risk to cooperate, which is 6, without exhausting his temptation to defect, which is 10. On the other hand, player 1 cannot preselect any defection-penalty that covers player 2's risk to cooperate, which is 7, without exhausting her temptation to defect, which is 5. It follows that player 1's conditional action in the prisoner's dilemma game cannot credibly deter player 2 from preselecting a defection-penalty, which makes (C; D) the only Nash equilibrium in stage 2 from which player 2 receives a higher payoff than what he would receive were they induced to cooperate. B. Necessary and Sufficient Conditions Suppose that a defection-penalty pair H Λ 2 H induces the players to cooperate. By Definition 1, there are action plans Φ Λ 1 and Φ Λ 2 such that Φ Λ 1(C; H Λ ) =Φ Λ 2(C; H Λ ) =1and 6

7 such that the strategy profile ((H1; Λ Φ Λ 1); (H2; Λ Φ Λ 2)) is a subgame-perfect equilibrium. From Figure 2, player 1 receives payoff R 1 and player 2 receives payoff R 2 from the subgameperfect equilibrium. Note first player 1 gains T 1 R 1 H Λ 1 units more by defecting in p (H Λ ), given that player 2 cooperates. Thus H Λ 1 must satisfy H Λ 1 T 1 R 1. Next, consider H 2 2 H 2 with H 2 < T 2 R 2. It must be Φ Λ 1(C; (H1;H Λ 2 )) < 1; otherwise player 2 receives payoff T 2 H 2 > R 2 by simply preselecting H 2 instead of H Λ 2 and by defecting in p (H1;H Λ 2 ), given player 1's strategy (H1; Λ Φ Λ 1). Consequently, given player 1's strategy (H1; Λ Φ Λ 1), by preselecting H 2 <T 2 R 2 and by cooperating in p (H1;H Λ 2 )player 2 gets expected payoff Φ Λ 1(H1;H Λ 2 )R 2 +[1 Φ Λ 1(H1;H Λ 2 )][S 2 + H1]. Λ Since Φ Λ 1(H1;H Λ 2 ) < 1 as argued above, player 1's defection-penalty H Λ 1 must satisfy H Λ 1» R 2 S 2, in order to make the above expected payoff for player 2nogreater than R 2. In summary, wehave shown T 1 R 1» H Λ 1» R 2 S 2. Similarly, T 2 R 2» H Λ 2» R 1 S 1. We have thus proved: LEMMA 1: Suppose that a defection-penalty pair H Λ 2Hinduces the players to cooperate. Then, for i 6= j, T i R i» H Λ i» R j S j. When P 1 S 1 < T 2 R 2, H Λ must satisfy some additional conditions. To see this, consider H 2 2H 2 with P 1 S 1 <H 2 <T 2 R 2. Then from Figure 2, H 2 >P 1 S 1 implies that the only optimal choice for player 1 in p (H1;H Λ 2 ) is to cooperate, given that player 2 defects. If H Λ 1 > T 1 R 1, then it is still the only optimal choice in p (H1;H Λ 2 ) for player 1 to cooperate, given that player 2 cooperates. Thus, together, H Λ 1 >T 1 R 1 and H 2 >P 1 S 1 imply that C is strictly dominant for player 1 in p (H1;H Λ 2 ). Consequently, since H 2 < T 2 R 2 and since Φ Λ (H1;H Λ 2 ) is a Nash equilibrium for p (H1;H Λ 2 ), Φ Λ 1(C; (H1;H Λ 2 )) = 1 and Φ Λ 2(C; (H1;H Λ 2 )) = 0. This shows that, when H Λ 1 >T 1 R 1,by deviating from (H2; Λ Φ Λ 2) to (H 2 ; Φ Λ 2) player 2 can guarantee himself payoff T 2 H 2 > R 2, which contradicts the fact that H Λ induces the players to cooperate. We conclude H Λ 1» T 1 R 1 whenever P 1 S 1 <T 2 R 2. If H Λ 1 <P 2 S 2, then this together with H 2 <T 2 R 2 implies that action D is strictly dominant for player 2 in p (H1;H Λ 2 ). Since H 2 >P 1 S 1 and since Φ Λ (H1;H Λ 2 ) is a Nash equilibrium for p (H1;H Λ 2 ), it must be Φ Λ 1(C; (H1;H Λ 2 )) = 1 and Φ Λ 2(C; (H1;H Λ 2 )) = 0. Thus, when H Λ 1 <P 2 S 2, by deviating from (H2; Λ Φ Λ 2) to (H 2 ; Φ Λ 2) player 2 can again guarantee himself payoff T 2 H 2 >R 2. This shows H Λ 1 P 2 S 2 whenever P 1 S 1 <T 2 R 2. To summarize, we have shown P 2 S 2» H Λ 1» T 1 R 1 whenever P 1 S 1 <T 2 R 2. By analogy, P 1 S 1» H Λ 2» T 2 R 2 whenever P 2 S 2 <T 1 R 1. We have thus proved: 7

8 LEMMA 2: Suppose that a defection-penalty pair H Λ 2Hinduces the players to cooperate. Then, for i 6= j, P j S j» H Λ i» T i R i whenever P i S i <T j R j. Conditions in Lemmas 1 and 2 turn out to be not only necessary but also sufficient for a defection-penalty pair H Λ to induce the players to cooperate. This result is summarized in the following theorem. THEOREM 1: A defection-penalty pair H Λ induces the players to cooperate if and only if, for i 6= j, T i R i» H Λ i» R j S j ; (1) and PROOF: See the Appendix. P j S j» H Λ i» T i R i whenever P i S i <T j R j : (2) When P 2 S 2 T 1 R 1 and P 1 S 1 T 2 R 2, the set of defection-penalty pairs inducing the players to cooperate is determined by condition (1) only. In this case, the set is rectangular. Figure 5 below provides such an example. [Insert FIGURE 5 here] The lower bound T 1 R 1 on defection-penalty H Λ 1 is needed to deter player 1 herself from defecting. On the other hand, the upper bound R 2 S 2 on H Λ 1 is needed for player 1 to deter player 2 from preselecting any defection-penalty smaller than T 2 R 2. Player 1's action to defect conditional on defection-penalties from player 2 smaller than T 2 R 2 provide the needed deterrence. However, such deterrence is not credible when H Λ 1 >R 2 S 2, because with H Λ 1 > R 2 S 2 player 2 would rather have player 1 defect, in which case he receives S 2 + H Λ 1 by cooperating himself, than have both player 1 and himself cooperate, in which case he receives R 2 <S 2 + H Λ 1 (see Figure 2). A similar intuitive explanation can be provided for the necessity of the lower bound T 2 R 2 and the upper bound R 1 S 1 on defection-penalty H2. Λ Note also that the compatibility of the upper and the lower bounds imply that mutual cooperation is efficient, in that sense that it gives rise to the highest total payoff (i.e., T 1 + S 2» R 1 + R 2 and T 2 + S 1» R 1 + R 2 ). Suppose P 1 S 1 < T 2 R 2. Then regardless of the defection-penalty H Λ 1 player 1 preselects, a defection-penalty H 2 from player 2 with P 1 S 1 <H 2 <T 2 R 2 is small enough to make it tempting for player 2 to defect in p (H 1 ;H 2 ) given that player 1 cooperates (i.e., R 2 <T 2 H 2 ), but large enough to make it the only optimal choice in p (H1;H Λ 2 ) for player 1 to cooperate given that player 2 defects (i.e., S 1 + H 2 >P 1 ). In this case, for player 1to 8

9 credibly deter player 2 from preselecting any such small defection-penalty by planning to defect conditional on such small defection-penalties from player 2, it would require that the defection-penalty H Λ 1 player 1 preselects be small enough for her to defect in p (H1;H Λ 2 ) given that player 2 cooperates (i.e., H Λ 1» T 1 R 1 ), but large enough for player 2 to cooperate, given that player 1 defects (i.e., H Λ 1 P 2 S 2 ). C. A Variant of the Penalty Mechanism Consider a variant of the penalty mechanism under which each player pays his or her preselected defection-penalty to the other player whenever he or she defects. Denote by p0 (H) the subgame in stage 2 following defection-penalty pair H. Then p0 (H) is given by: [Insert FIGURE 6 here] It turns out that under this variant a defection-penalty pair H Λ induces the players to cooperate if and only if T i R i» H Λ i» minfp i S i ;R j P j g, for i 6= j. To see this, let H Λ be a defection-penalty pair inducing the players to cooperate under the variant of the penalty mechanism. By Definition 1, there are action plans Φ Λ 1 and Φ Λ 2 such that Φ Λ 1(C; H Λ ) = Φ Λ 2(C; H Λ ) = 1 and such that the strategy profile ((H1; Λ Φ Λ 1); (H2; Λ Φ Λ 2)) is a subgame-perfect equilibrium. Player 1 receives payoff R 1 and player 2 receives payoff R 2 from the subgame-perfect equilibrium. From Figure 6, player 1 gains T i R i H Λ i units more by defecting in p0 (H Λ ), given that player 2 cooperates. It follows H Λ 1 T 1 R 1. Now consider defection-penalty H 2 = 0 from player 2. From Figure 6, action D is strictly dominant for player 2 in (H1;H p0 Λ 2 ). Suppose H Λ 1 > P 1 S 1. Then, the only optimal choice for player 1 in p0 (H1;H Λ 2 ) is to cooperate, given that player 2 defects. It follows that the only Nash equilibrium for (H1;H p0 Λ 2 ) is (C; D). Consequently, given player 1's strategy (H1; Λ Φ Λ 1), player 2 receives payoff T 2 > R 2 by deviating from (H2; Λ Φ Λ 2) to (H 2 ; Φ Λ 2), which contradicts the fact that H Λ induces the players to cooperate. This shows H Λ 1» P 1 S 1 and Φ Λ 1(C; (H1;H Λ 2 )) < 1. Thus given player 1's strategy (H1; Λ Φ Λ 1), by preselecting penalty H 2 =0and by defecting in p0 (H1;H Λ 2 ) player 2 gets expected payoff Φ Λ 1(C; (H1;H Λ 2 ))(T 2 H 2 )+Φ Λ 1(D; (H1;H Λ 2 ))[P 2 H 2 + H1]. Λ Since Φ Λ 1(C; (H1;H Λ 2 )) < 1 and since H 2 =0,for this expected payoff to be no greater than R 2, it must be P 2 + H Λ 1» R 2 or equivalently H Λ 1» R 2 P 2. In summary, wehaveshown that H Λ 1 must satisfy T 1 R 1» H Λ 1» minfp 1 S 1 ; R 2 P 2 g. By analogy, H Λ 2 must satisfy T 2 R 2» H Λ 2» minfp 2 S 2 ; R 1 P 1 g. Conversely, let H Λ be a defection-penalty pair satisfying T 1 R 1» H Λ 1» minfp 1 S 1 ; R 2 P 2 g and T 2 R 2» H Λ 2» minfp 2 S 2 ; R 1 P 1 g. For H 2 2H 2, let Φ Λ 1(H1;H Λ 2 ) 9

10 and Φ Λ 2(H1;H Λ 2 ) be defined by Φ Λ 1(D; (H1;H Λ 2 )) =1 Φ Λ 1(C; (H1;H Λ 2 )), Φ Λ 2(D; (H1;H Λ 2 )) = 1 Φ Λ 2(C; (H1;H Λ 2 )), and Φ Λ 1(C; (H Λ 1;H 2 )) = Φ Λ 2(C; (H Λ 1;H 2 )) = ( 1 if H2 T 2 R 2 ; 0 if H 2 <T 2 R 2 : (3) For H 1 2 H 1, let Φ Λ 1(H 1 ;H2) Λ and Φ Λ 2(H 1 ;H2) Λ be defined analogously. Finally, for H 2 H with H 1 6= H Λ 1 and H 2 6= H2, Λ let Φ Λ (H) beany Nash equilibrium for the subgame p0 (H). Note that, since H Λ 1 T 1 R 1 and H Λ 2 T 2 R 2, (3) implies Φ Λ 1(C; H Λ )=Φ Λ 2(C; H Λ )= 1. Now consider any defection-penalty H 2 2 H 2. Observe first that the conditions on H Λ imply T 1 R 1» H Λ 1» P 1 S 1 and T 2 R 2» H Λ 2» P 2 S 2. Thus, (C; C) is a Nash equilibrium for p0 (H1;H Λ 2 ) when H 2 T 2 R 2 ; (D; D) is a Nash equilibrium for (H1;H p0 Λ 2 ) when H 2 < T 2 R 2. This shows that for any H 2 2 H 2, the action pair Φ Λ (H1;H Λ 2 ) as in (3) is a Nash equilibrium for (H1;H p0 Λ 2 ). By analogy, for any H 1 2H 1, the action pair Φ Λ (H 1 ;H2) Λ is a Nash equilibrium for p0 (H 1 ;H2). Λ Thus to show that H Λ induces the players to cooperate, it only remains to prove that the players do not have any incentive to unilaterally change their defection-penalties in H Λ. To this end, consider any defection-penalty H 2 2 H 2. By (3) and Figure 6, player 2's payoff at strategy profile ((H1; Λ Φ Λ 1); (H 2 ; Φ Λ 2)) is R 2 when H 2 T 2 R 2 and his payoff is P 2 H 2 + H Λ 1 when H 2 < T 2 R 2. Since H Λ 1» R 2 P 2 and since player 2's payoff at strategy profile ((H1; Λ Φ Λ 1); (H2; Λ Φ Λ 2)) is R 2, player 2 has no incentive to unilaterally change his defection-penalty H2. Λ By analogy, player 1 has no incentive to unilaterally change her defection-penalty H1. Λ We have therefore proved: THEOREM 1 0 : A defection-penalty pair H Λ variant of the penalty mechanism if and only if, for i 6= j, induces the players to cooperate under the T i R i» H Λ i» minfp i S i ; R j P j g: (1 0 ) Since S 1 <P 1 and S 2 <P 2, (1 0 ) implies T 1 R 1» H Λ 1 <R 2 S 2 and T 2 R 2» H Λ 2 < R 1 S 1. This means that condition (1 0 ) is more restrictive than condition (1). However, condition (2) is not required for a defection-penalty pair to induce the players to cooperate under the variant of the penalty mechanism. Thus, (1) and (2) do not necessarily imply (1 0 ) nor conversely. Nevertheless, the prisoner's dilemma game in Figure 3 fails to satisfy condition (1 0 ). Consequently, the variant of the penalty mechanism also fails to promote cooperation in that game. We show in Section 4 that condition (1 0 ) imposes stronger conditions on the payoffs for prisoner's dilemma than those necessary and sufficient for cooperation to be inducible via cooperation-rewards. 10

11 D. An Application to Law and Economics In the law of contracts, an expectation damage remedy" is defined as a payment that places the victim of a breached contract in the position he or she would have been in had the other party performed (Cooter and Ulen 2000, pp. 226). From Figure 1, if player 1 cooperates and player 2 defects, player 1's payoff is S 1. Player 1's payoff would have been R 1 if player 2 had cooperated. Thus to place player 1 in the position she would have been in had player 2 cooperated, it would require that player 2 pay player 1 the amount R 1 S 1. It follows that the expectation damage remedy that player 2 pays to player 1 when he is held liable for expectation damages is R 1 S 1. Similarly, if player 1 defects while player 2 cooperates, the expectation damage remedy that player 1 pays to player 2 when she is held liable for expectation damages is R 2 S 2. A disgorgement remedy" is a payment paid to the victim of a breached contract to eliminate the breacher's profit from wrong doing (Cooter and Ulen (2000, pp. 234). From Figure 1, player 2 gains T 2 R 2 units more from defecting given that player 1 cooperates. Thus to eliminate this gain from wrong doing (defecting), it would require that player 2 pay player 1 the amount equal to his temptation to defect, T 2 R 2. It follows that the disgorgement damage remedy that player 2 pays to player 1 when he is held liable for disgorgement damages is T 2 R 2. Similarly, if player 1 defects while player 2 cooperates, the disgorgement damage remedy that player 1 pays to player 2 when she is held liable for disgorgement damages is T 1 R 1. Finally, a liquidated damage remedy" is one under which, if one party breaches, the victim can recover an amount predetermined by the parties to a contract. (Cooter and Ulen 2000, pp ). 5 Liquidated damage remedies differ from the other two, in that they are party-designed rather than imposed upon the parties by a third-party such as a court. Defection-penalties are payments from defectors to cooperators which are preselected by the players themselves. As such, defection-penalties are liquidated damage remedies that defectors pay to cooperators. When P 1 S 1 <T 2 R 2 and P 2 S 2 <T 1 R 1, Theorem 1 implies that to induce cooperation it is necessary and sufficient for each player to pay the other player a defection-penalty, which is equal to what the player pays to the other player when held liable for disgorgement damages. When P 1 S 1 T 2 R 2 and P 2 S 2 T 1 R 1, Theorem 1 implies that to induce cooperation it is necessary and sufficient for each player to pay the other player a defection-penalty bounded below and above respectively by the disgorgement and the expectation damage remedies, which the player pays to the other 5 In the law ofcontracts, this label is used only for those parties' stipulated damage remedies that are deemed legally enforceable. I use the term here to refer to any damage remedy the parties may stipulate. 11

12 player when held liable for these damages. A preselected defection-penalty from a player, which is larger than the expectation damage remedy the player pays when held liable for expectation damages, cannot make it in the other player's best interests to preselect a defection-penalty so as to induce both players to cooperate, regardless of whether mutual cooperation is efficient or not. On that ground, as a liquidated damage remedy such a defection-penalty should be regarded as unenforceable Promoting Cooperation via the Reward Mechanism Under the reward mechanism a player gets compensated for cooperation with a preselected cooperation-reward from the other player. We let H 1 and H 2 now denote the sets of cooperation-rewards player 1 and player 2 can respectively preselect. We also assume H 1 =[0; 1) andh 2 =[0; 1). As with the penaltymechanism, a strategy for a player with the reward mechanism must specify a cooperation-reward that the player will preselect and in addition, the strategy must also specify an action plan that determines how the player's action in prisoner's dilemma reacts to the various cooperation-rewards they each may preselect. We use the same notation (H 1 ; Φ 1 ) to denote a generic strategy for player 1 and (H 2 ; Φ 2 ) a generic strategy for player 2. A cooperation-reward pair H changes the prisoner's dilemma game in Figure 2 into game r (H) in Figure 7 below, which is the subgame following cooperation-reward pair H. 6 [Insert FIGURE 7 here] We consider the possibility for inducing the players to cooperate via cooperation-rewards in subgame-perfect equilibrium: DEFINITION 2: A cooperation-reward pair H Λ =(H Λ 1;H Λ 2) induces the players to cooperate if there are action plans Φ Λ 1 and ΦΛ 2 such thatφλ 1(C; H Λ )=Φ Λ 2(C; H Λ )=1and such that the strategy profile ((H Λ 1; Φ Λ 1); (H Λ 2; Φ Λ 2)) is a subgame-perfect equilibrium for the two-stage game. That is, a cooperation-reward pair H Λ = (H Λ 1;H Λ 2) induces the players to cooperate if, as a subgame-perfect equilibrium, player 1 selects H Λ 1 and player 2 selects H Λ 2 in stage 1 and they both cooperate in stage 2 conditional on these cooperation-rewards. We say that the 6 As before, when the cooperation-reward pair H is clearly understood, we will omit the phrase following cooperation-reward pair H". 12

13 reward mechanism promotes cooperation if there exists a cooperation-reward pair inducing the players to cooperate. As with the penalty mechanism, we assume that the players can select any nonnegative cooperation-rewards. That is, we assume H 1 = [0; 1) and H 2 =[0; 1). As Figure 7 shows, the players give each other their preselected cooperation-rewards in the event ofmutual cooperation. This means that the larger a cooperation-reward a player preselects, the more costly it will be for the player to induce the other player to cooperate. Therefore, cooperation-rewards inducing the players to cooperate must be minimal, inthe sense that neither player can reduce his or her preselected cooperation-reward and still induces the other player to cooperate. A. An Example Cooperation is not always inducible via cooperation-rewards. To see this, note that with the prisoner's dilemma game in Figures 3, a cooperation-reward pair H leads to the following subgame r (H): [Insert FIGURE 8 here] First, player 2's temptation to defect in the prisoner's dilemma game in Figure 3 is 10. Thus, to induce player 2 to cooperate, player 1 must give him a cooperation-reward H Λ 1 10; otherwise he would be better off defecting given that player 1 cooperates (i.e. 20 H Λ 2 + H Λ 1 < 30 H Λ 2 whenever H Λ 1 < 10). Second, player 1's temptation to defect in the prisoner's dilemma game in Figure 3 is 5. Thus, player 2 must give player 1 a cooperation-reward H Λ 2 5 to induce her to cooperate. From Figure 8, with H Λ 1 10 any cooperation-reward H 2 from player 2 with 5 <H 2 < 6 induces player 1 to cooperate in any Nash equilibrium for r (H1;H Λ 2 ). 7 This implies that if the players were induced to cooperate, player 2 must give player 1 cooperation-reward H Λ 2 = 5 to minimize the cost for inducing her cooperation. Thus, player 1 receives 20 H Λ 1» 10 were they induced to cooperate. 7 Suppose first H Λ 1 > 10. Then, in this case, action C is strictly dominant for player 2 in r (H Λ 1 ;H 2). This together with H 2 > 5 implies that (C; C) is the only Nash equilibrium for r (H Λ 1 ;H 2). Suppose now H Λ 1 =10. In this case, 5 <H 2 < 6 implies that player 1 is better off playing D conditional on player 2 also playing D; player 2 is better of playing C conditional on player 1 playing D; player 1 is better off playing C conditional on player 2 also playing C; and finally, player 2 is indifferent between playing C and playing D conditional on player 1 playing C. Thus, action C is the only best response for player 2 whenever player 1 plays action D with positive probability. Since action C is the only best response for player 1 to player 2's action C, there cannot be any Nash equilibrium for r (H Λ 1 ;H 2) with player 1 playing action D with positive probability. 13

14 Third, with cooperation-reward H Λ 2 = 5 from player 2, action D is dominant for player 1 in r (H 1 ;H Λ 2), for any H 1 2 H 1. 8 Consequently, by similar reasoning as in footnote 7, any cooperation-reward H 1 with 7 < H 1 < 10 induces player 2 to cooperate in any Nash equilibrium for r (H 1 ;H Λ 2). Hence, given any subgame-perfect equilibrium strategy (H Λ 2; Φ Λ 2) with H Λ 2 = 5 for player 2, by selecting a cooperation-reward H 1 with 7 <H 1 < 10 in stage 1 and by defecting in stage 2 conditional on (H 1 ;H Λ 2), player 1 guarantees herself payoff 20 H 1 > 10. As a result, no cooperation-reward can induce the players to cooperate because, as shown above, player 1 receives no more than 10 were they induced to cooperate. B. Necessary and Sufficient Conditions Let H Λ 2Hbe a cooperation-reward pair inducing the players to cooperate. By Definition 2, there are action plans Φ Λ 1 and Φ Λ 2 such that Φ Λ 1(C; H Λ ) = Φ Λ 2(C; H Λ ) = 1 and such that the strategy profile ((H1; Λ Φ Λ 1); (H2; Λ Φ Λ 2)) is a subgame perfect equilibrium. Player 1 receives payoff R 1 H Λ 1 + H Λ 2 and player 2 receives payoff R 2 H Λ 2 + H Λ 1 from the subgame-perfect equilibrium. From Figure 7, player 1 gains T 1 R 1 H Λ 2 units more by defecting in in r (H Λ ), given that player 2 cooperates. Thus, H Λ 2 T 1 R 1. Similarly, H Λ 1 T 2 R 2. Next, let H 2 2H 2 be such that H 2 < minft 1 R 1 ;P 1 S 1 g. Then action D is strictly dominant for player 1 in r (H1;H Λ 2 ). Since Φ Λ (H1;H Λ 2 ) is a Nash equilibrium for r (H1;H Λ 2 ), Φ Λ 1(C; (H1;H Λ 2 )) =0. Consequently, given player 1's strategy (H1; Λ Φ Λ 1), by preselecting H 2 player 2 receives payoff S 2 + H Λ 1 when he cooperates in r (H1;H Λ 2 ) and he receives payoff P 2 when he defects in r (H1;H Λ 2 ). We conclude that H Λ must satisfy R 2 H Λ 2 +H Λ 1 S 2 +H Λ 1 and R 2 H Λ 2 +H Λ 1 P 2. This shows H Λ 2» R 2 S 2 and H Λ 2 H Λ 1» R 2 P 2. Similarly, H Λ 1» R 1 S 1 and H Λ 1 H Λ 2» R 1 P 1. We have thus proved: LEMMA 3: Suppose that a cooperation-reward pair H Λ 2Hinduces the players to cooperate. Then, for i 6= j, T i R i» H Λ j» R j S j and H Λ j H Λ i» R j P j. Mutual cooperation is not always inducible by cooperation-reward pairs satisfying conditions in Lemma 3 alone. For example, when P 1 S 1 > T 1 R 1, cooperation-rewards H Λ 1 and H Λ 2 with H Λ 1 > P 2 S 2 and H Λ 2 > P 1 S 1 are too large to induce the players to cooperate. In this case, some further conditions are required. Theorem 2 below summarizes necessary and sufficient conditions for a cooperation-reward pair to induce the players to cooperate. 8 For the rest of the paper, we use the term dominance to mean either the usual strict or the usual weak dominance. 14

15 THEOREM 2: A cooperation-reward pair H Λ 2Hinduces the players to cooperate if and only if, for i 6= j, T i R i» H Λ j» R j S j ; (4) and PROOF: See the Appendix. H Λ j H Λ i» R j P j ; (5) H Λ j» T i R i whenever P i S i» T i R i ; (6) H Λ i» P j S j and H Λ j» P i S i whenever P i S i >T i R i : (7) When P 1 S 1 >T 1 R 1 and P 2 S 2 >T 2 R 2, the set of cooperation-reward pairs inducing the players to cooperate is determined by conditions (4), (5), and (7) only. In this case, the set is sometimes an hexagon. Figure 9 below provides such an example. [Insert FIGURE 9 here] Condition (4) implies T 1 + S 2» R 1 + R 2 and T 2 + S 1» R 1 + R 2. This means that mutual cooperation must be efficient for it to be inducible. When H Λ 1 < T 2 R 2, the cooperation-reward is not large enough to induce player 2 to cooperate. On the other hand, when H Λ 1 > R 1 S 1, the cooperation-reward is so large that player 1 would rather have player 2 defect, in which case she receives payoff S 1 + H Λ 2 by cooperating, than have both of them cooperate, in which case she receives payoff R 1 H Λ 1 + H Λ 2 <S 1 + H2. Λ When H Λ 1 H Λ 2 >R 1 P 1, cooperation-reward H Λ 1 is so large relative to player 2's cooperationreward H Λ 2 to player 1 that she prefers mutual defection, from which she receives payoff P 1, to mutual cooperation, from which she receives payoff R 1 H Λ 1 + H Λ 2 <P 1. This explains the necessity of (4) and (5) for the case where i =2and j =1. The case with i =1and j = 2canbe explained analogously. When H Λ 2 > maxfp 1 S 1 ;T 1 R 1 g, action C is strictly dominant for player 1 in r (H 1 ;H2), Λ H 1 2H 1. In this case, player 2 can reduce his cooperation-reward H Λ 2 and still induces player 1 to cooperate. Consequently, it must be H Λ 2» maxfp 1 S 1 ;T 1 R 1 g. By analogy, H Λ 1» maxfp 2 S 2 ;T 2 R 2 g. Thus, when P 1 S 1 >T 1 R 1, H Λ 1 >P 2 S 2 implies that player 2's risk to cooperate, P 2 S 2, is less than his temptation to defect, T 2 R 2, while the opposite holds for player 1. When P 2 S 2 > T 2 R 2, H Λ 2 > P 1 S 1 implies that player 1's risk to cooperate, P 1 S 1, is less than her temptation to defect, T 1 R 1, while the opposite holds for player 2. As the example in Section 3.2.A demonstrates, no 15

16 cooperation-reward pair induces the players to cooperate in either of these two cases. 9 Thus H Λ 1» P 2 S 2 when P 1 S 1 >T 1 R 1 and H Λ 2» P 1 S 1 when P 2 S 2 >T 2 R 2. This explains the necessity of (6) and (7). C. Cooperation-Rewards as Coasian Payments Coase (1960) establishes the result known as the Coase Theorem by considering, among other examples, a rancher and a farmer with adjoining properties. The rancher's cattle occasionally stray onto the farmer's property and destroy some of the crops. The Coase Theorem implies that even if the rancher were not liable for damages, an efficient outcome would still result, because the farmer then has an incentive to offer to pay the rancher to reduce the number of straying cattle (i.e. reward him for cooperation). Cooperationrewards compensate the players for taking the initially less desirable action. As such, they are payments as considered in Coase (1960). Theorem 2 characterizes such Coasian payments that are necessary and sufficient for inducing the players to cooperate in prisoner's dilemma. As shown in next section, Theorem 2 implies necessary and sufficient conditions on the payoffs for cooperation to be inducible with such Coasian payments. 4. Penalties versus Rewards as Inducements to Cooperate This section concerns a partition of the payoffs for prisoner's dilemma according to whether both the penalty mechanism and the reward mechanism promote cooperation, one but not both mechanisms promotes cooperation, and neither of the two mechanisms promotes cooperation. 10 To this end, we first establish necessary and sufficient conditions on the payoffs respectively for the penalty and the reward mechanisms to promote cooperation. 9 Recall that in that example, player 1's temptation to defect is 5 and her risk to cooperate is 6, whereas player 2's temptation to defect and risk to cooperate are respectively 10 and 7. That cooperation in this case is not inducible via cooperation-rewards also follows from Proposition 1 of Ziss (1997). See Lemma 4 in the Appendix of the present paper. 10 Cooperation-rewards and defection-penalties generate strategy dependent payoff transfers between the players. Jackson and Wilkie (2002) also considers strategy dependent payoff transfers between the players in strategic games. Payoff transfers in that paper are not restricted to be achievable through defectionpenalties only or through cooperation-rewards only. For example, with a prisoner's dilemma game players in Jackson and Wilkie (2002) can prearrange, among the permissible strategy dependent payoff transfers, those payoff transfers which will be carried out only when both players defect. We consider cooperationrewards only or defection-penalties only as means of transferring payoffs between the players because of their respective relations to Coasian payments and liquidated damage remedies. 16

17 THEOREM 3: Necessary and sufficient conditions for the penalty mechanism, its variant, and for the reward mechanism to promote cooperation are respectively and (T 1 R 1 + S 2 P 2 )(T 2 R 2 + S 1 P 1 ) 0; T i R i» R j S j for i 6= j; (8) T i R i» minfp i S i ; R j P j g for i 6= j; (9) (T 1 R 1 + S 1 P 1 )(T 2 R 2 + S 2 P 2 ) 0; T i R i» R j S j for i 6= j: (10) PROOF: The necessity of (8) follows directly from (1) and (2). For the sufficiency, let H Λ 1 = T 1 R 1 and H Λ 2 = T 2 R 2. Then, with (8) the defection-penalty pair H Λ =(H Λ 1;H Λ 2) satisfies (1) and (2). By Theorem 1, H Λ induces the players to cooperate. The necessity of (9) follows directly from (1 0 ). For the sufficiency, set H Λ 1 = T 1 R 1 and H Λ 2 = T 2 R 2. It then follows from (9) that the defection-penalty pair H Λ =(H Λ 1;H Λ 2) satisfies (1 0 ). By Theorem 1 0, H Λ induces the players to cooperate. The necessity of (10) are implied by (4)-(7). To show the sufficiency, observe first that (10) implies either P 1 S 1» T 1 R 1 and P 2 S 2» T 2 R 2 or P 1 S 1 T 1 R 1 and P 2 S 2 T 2 R 2. Now consider cooperation-rewards H Λ 1 and H Λ 2, where H Λ 1 = T 2 R 2 and H Λ 2 = T 1 R 1 when P 1 S 1» T 1 R 1 and P 2 S 2» T 2 R 2 ; H Λ 1 = maxft 2 R 2 ; (T 1 R 1 ) (R 2 P 2 )g and H Λ 2 = T 1 R 1 when P 1 S 1 = T 1 R 1 and P 2 S 2 >T 2 R 2 ; and H Λ 1 = maxft 2 R 2 ; (T 1 R 1 ) (R 2 P 2 )g and H Λ 2 =minfh Λ 1 + R 2 P 2 ;P 1 S 1 g when P 1 S 1 >T 1 R 1 and P 2 S 2 T 2 R 2. Assume first P 1 S 1» T 1 R 1 and P 2 S 2» T 2 R 2. In this case, (7) is irrelevant. By construction, H Λ 1 H Λ 2» R 1 P 1 if and only if T 2 R 2» T 1 P 1. Since T 2 R 2» R 1 S 1 as in (10) and since R 1 S 1» T 1 P 1 as implied by assumption, we have T 2 R 2» T 1 P 1. Hence H Λ 1 H Λ 2» R 1 P 1. Similarly, H Λ 2 H Λ 1» R 2 P 2. This shows that H Λ satisfies (5). By construction and by (10), H Λ also satisfies (4) and (6). Assume now P 1 S 1 = T 1 R 1 and P 2 S 2 > T 2 R 2. In this case, note first (T 1 R 1 ) (R 2 P 2 )» P 2 S 2 if and only if T 1 R 1» R 2 S 2 which, by (10), is satisfied. This together with the assumed inequality T 2 R 2 < P 2 S 2 implies H Λ 1» P 2 S 2. On the other hand, T 1 R 1 = P 1 S 1 < R 1 S 1. Hence by (10) and by construction, T 2 R 2» H Λ 1» R 1 S 1, T 1 R 1» H Λ 2» R 2 S 2, and H Λ 2» P 1 S 1. This shows H Λ satisfies (4), (6), and (7). Since H Λ 1 (T 1 R 1 ) (R 2 P 2 ) and H Λ 2 = T 1 R 1, we have H Λ 2 H Λ 1» R 2 P 2. On the other hand, H Λ 1 H Λ 2» R 1 P 1 if and only if T 2 R 2» T 1 P 1. Since T 1 P 1 = R 1 S 1 as implied by assumption and since T 2 R 2» R 1 S 1 as in (10), we have H Λ 1 H Λ 2» R 1 P 1. This that H Λ also satisfies (5). 17

18 Assume finally, P 1 S 1 > T 1 R 1 and P 2 S 2 T 2 R 2. In this case, note first T 2 R 2» H Λ 1» R 1 S 1 and H Λ 1» P 2 S 2 as in the previous case. Note also H Λ 2» P 1 S 1 by construction. Since P 1 S 1 >T 1 R 1,wehave H Λ 2 = T 1 R 1 if H Λ 1 =(T 1 R 1 ) (R 2 P 2 ) and H Λ 2 = minft 2 P 2 ;P 1 S 1 g if H Λ 1 = T 2 R 2. On the other hand, H Λ 1 = T 2 R 2 if and only if T 2 P 2 T 1 R 1. Thus, by (10) and by the inequalities P 1 S 1 > T 1 R 1 and T 2 P 2» R 2 S 2 as implied by assumption, T 1 R 1» H Λ 2» R 2 S 2. We conclude that H Λ satisfies (4), (6), and (7). Next, H Λ 1 H Λ 2 = (R 2 P 2 )ifh Λ 1 =(T 1 R 1 ) (R 2 P 2 ) and H Λ 1 H Λ 2 =maxfp 2 R 2 ; (T 2 R 2 ) (P 1 S 1 )g if H Λ 1 = T 2 R 2. Since T 2 R 2» R 1 S 1 as in (10), we have H Λ 1 H Λ 2» R 1 P 1. Furthermore, since H Λ 2» H Λ 1 + R 2 P 2, we have H Λ 2 H Λ 1» R 2 P 2. This shows H Λ also satisfies (5). To summarize, we have shown that with condition (10) the cooperation-reward pair H Λ as constructed above satisfies (4)-(7). Hence, by Theorem 2, H Λ induces the players to cooperate. In words, (8) states that mutual cooperation is efficient and that either each player's temptation to defect is no less than the other player's risk to cooperate, or each player's temptation to defect is no greater than the other player's risk to cooperate. Condition (9) states that each player's temptation to defect is no greater than either his or her own risk to cooperate or the other player's gain for mutual cooperation. Finally, condition (10) states that mutual cooperation is efficient and that either each player's temptation to defect is no less than his or her own risk to cooperate, or each player's temptation to defect is no greater than his or her own risk to cooperate. Conditions (8) and (10) together imply the following comparison result: First, both the penalty mechanism and the reward mechanism promote cooperation when mutual cooperation is efficient, (T 1 R 1 +S 2 P 2 )(T 2 R 2 +S 1 P 1 ) 0, and (T 1 R 1 +S 1 P 1 )(T 2 R 2 +S 2 P 2 ) 0. Second, the penalty mechanism promotes cooperation but the reward mechanism does not when mutual cooperation is efficient, (T 1 R 1 + S 2 P 2 )(T 2 R 2 + S 1 P 1 ) 0, and (T 1 R 1 + S 1 P 1 )(T 2 R 2 + S 2 P 2 ) < 0. Third, the reward mechanism promotes cooperation but the penalty mechanism does not when mutual cooperation is efficient, (T 1 R 1 + S 2 P 2 )(T 2 R 2 + S 1 P 1 ) < 0, and (T 1 R 1 + S 1 P 1 )(T 2 R 2 + S 2 P 2 ) 0. Fourth, neither the penalty nor the reward mechanism promotes cooperation when either mutual cooperation is inefficient or (T 1 R 1 + S 2 P 2 )(T 2 R 2 + S 1 P 1 ) < 0 and (T 1 R 1 + S 1 P 1 )(T 2 R 2 + S 2 P 2 ) < 0. The above four classes of the payoffs are mutually exclusive and jointly exhaustive. Hence, they constitute a partition of the payoffs for prisoner's dilemma. Note also that, since R 1 P 1 <R 1 S 1 and R 2 P 2 <R 2 S 2, (10) is satisfied whenever (9) is satisfied. It follows that the reward mechanism promotes cooperation whenever the 18

19 variant of the penalty mechanism does. 5. Summary In this paper we considered the possibility for inducing the players to cooperate in prisoner's dilemma via cooperation-rewards only or via defection-penalties only. We completely characterized cooperation-rewards and defection-penalties inducing the players to cooperate. Our characterization of cooperation-rewards inducing the players to cooperate implies that, for mutual cooperation to be inducible via cooperation-rewards only, it is necessary and sufficient that mutual cooperation is efficient and that either each player's temptation to defect is no less thanhisorherown risk to cooperate, or each player's temptation is no greater than his or her own risk to cooperate. On the other hand, our characterization of defection-penalties inducing the players to cooperate implies that, for mutual cooperation to be inducible via defection-penalties only, it is necessary and sufficient that mutual cooperation is efficient and that either each player's temptation to defect is no less than the other player's risk to cooperate, or each player's temptation to defect is no greater than the other player's risk to cooperate. Appendix: PROOFS Let U 1 ((H 1 ; Φ 1 ); (H 2 ; Φ 2 )) and U 2 ((H 1 ; Φ 1 ); (H 2 ; Φ 2 )) denote respectively the payoffs for player 1 and player 2 at strategy profile ((H 1 ; Φ 1 ); (H 2 ; Φ 2 )) under either the penalty orthe reward mechanism. They are the payoffs at action pair (Φ 1 (H); Φ 2 (H)) in p (H) with the penalty mechanism or in r (H) with the reward mechanism. PROOF OF THEOREM 1: The necessity of (1) and (2) follows from Lemma 1 and Lemma 2. Thus it only remains to prove the sufficiency of these conditions. Let H Λ 2Hbe a penalty pair satisfying conditions (1)-(2). For H 2 2H 2,letΦ Λ 1(H Λ 1;H 2 ) and Φ Λ 2(H Λ 1;H 2 ) be defined by Φ Λ 1(D; (H Λ 1;H 2 )) =1 Φ Λ 1(C; (H Λ 1;H 2 )), Φ Λ 2(D; (H Λ 1;H 2 )) = 1 Φ Λ 2(C; (H Λ 1;H 2 )), Φ Λ 1(C; (H Λ 1;H 2 )) = 8 >< >: 1 if H 2 T 2 R 2 ; 0 if P 1 S 1» H 2 <T 2 R 2 ; 0 if H Λ 1» P 2 S 2 ; H 2 < minfp 1 S 1 ;T 2 R 2 g; S 2 +H Λ 1 P 2 T 2 R 2 H 2 +S 2 +H Λ 1 P 2 if H Λ 1 >P 2 S 2 ; H 2 < minfp 1 S 1 ;T 2 R 2 g; (11) and 19

20 Φ Λ 2(C; (H Λ 1;H 2 )) = 8 >< >: 1 if H 2 T 2 R 2 ; 1 if P 1 S 1» H 2 <T 2 R 2 ; 0 if H Λ 1» P 2 S 2 ; H 2 < minfp 1 S 1 ;T 2 R 2 g; S 1 +H 2 P 1 T 1 R 1 H Λ 1 +S 1+H 2 P 1 if H Λ 1 >P 2 S 2 ; H 2 < minfp 1 S 1 ;T 2 R 2 g: (12) For H 1 2 H 1, let Φ Λ 1(H 1 ;H2) Λ and Φ Λ 2(H 1 ;H2) Λ be defined analogously. Finally, for H 2 H with H 1 6= H Λ 1 and H 2 6= H2, Λ let (Φ Λ 1(H); Φ Λ 2(H)) be any Nash equilibrium for the subgame p (H). Note first (1), (11), and (12) imply Φ Λ 1(C; H Λ )=Φ Λ 2(C; H Λ )=1. Now consider any defection-penalty H 2 2 H 2. By (1), H Λ 1 T 1 R 1. It follows that (C; C) is a Nash equilibrium for p (H1;H Λ 2 ) when H 2 T 2 R 2. Suppose P 1 S 1» H 2 < T 2 R 2. In this case, P 1 S 1 < T 2 R 2 and hence by (1) and (2), P 2 S 2» H Λ 1 and H Λ 1 = T 1 R 1. Consequently, (D; C) is a Nash equilibrium for p (H1;H Λ 2 ). Suppose now H Λ 1» P 2 S 2 and H 2 < minfp 1 S 1 ;T 2 R 2 g. In this case, S 1 +H 2 <P 1 and S 2 +H Λ 1» P 2 and so, (D; D) is a Nash equilibrium for p (H1;H Λ 2 ). Suppose finally H Λ 1 > P 2 S 2 and H 2 < minfp 1 S 1 ;T 2 R 2 g. In this case, (11) and (12) imply that given player 2's strategy (H 2 ; Φ Λ 2), action C and action D yield the same payoff to player 1 in p (H1;H Λ 2 ). Similarly, given player 1's strategy (H1; Λ Φ Λ 1), action C and action D yield the same payoff to player 2in p (H1;H Λ 2 ). Thus, Φ Λ (H1;H Λ 2 )isa Nash equilibrium for p (H1;H Λ 2 ). In summary, we have shown that for any H 2 2 H 2, Φ Λ (H1;H Λ 2 ) as in (11) and (12) is a Nash equilibrium for p (H1;H Λ 2 ). By analogy, for any H 1 2 H 1, Φ Λ (H 1 ;H2) Λ is a Nash equilibrium for p (H 1 ;H2). Λ Thus, to complete the proof of the sufficiency, it only remains to show that no player has any incentive to unilaterally change his or her penalty in H Λ. To this end, consider any defection-penalty H 2 2H 2. By (11), (12), and Figure 2, U 2 ((H Λ 1; Φ Λ 1); (H 2 ; Φ Λ 2)) = 8 >< >: R 2 if H 2 T 2 R 2 ; S 2 + H Λ 1 if P 1 S 1» H 2 <T 2 R 2 ; P 2 if H Λ 1» P 2 S 2 ; H 2 < minfp 1 S 1 ;T 2 R 2 g; R 0 2 if H Λ 1 >P 2 S 2 ; H 2 < minfp 1 S 1 ;T 2 R 2 g; (13) where R 0 2 =Φ Λ 1(C; (H1;H Λ 2 ))R 2 +Φ Λ 1(D; (H1;H Λ 2 ))(S 2 + H1). Λ By (1), H Λ 1» R 2 S 2 which implies R 0 2» R 2. Consequently, by (13), U 2 ((H1; Λ Φ Λ 1); (H 2 ; Φ Λ 2))» R 2. This shows that player 2 does not have any incentive to change his defection-penalty H2. Λ Similarly, player 1doesnot have any incentive to change her defection-penalty H1. Λ 20