period one to have external validity since we cannot apply them in our real life if it takes many periods to achieve the goal of them. In order to cop

Transcription

1 Second Thought: Theory and Experiment in Social ilemma Saijo, Tatsuyoshi and Okano, Yoshitaka (Kochitech) 1. Introduction Why have we been using second thought? This paper shows that second thought is not an innocent device in our daily life, but is human wisdom that plays an important role to resolve problems such as social dilemma. We design a simplest possible mechanism to achieve Pareto efficiency in social dilemma, and then compare the performance of the mechanism with and without second thought. The mechanism with second thought performs very well in theory and experiment. Moreover, the mechanism with second thought is robust against deviation from rational choices. Thus, incorporating wisdom into mechanism design is a new powerful tool. Let us start with designing mechanisms taming social dilemma. Saijo, Okano and Yamakawa (2013) designed a mechanism called the approval mechanism (AM) that implements cooperation in prisoner's dilemma (P) game and then conducted the experiments. In theory, the AM works well, and in experiment, the cooperation rate of the AM without repetition is 100% and it is more than 90% in all periods with repetition. Furthermore, the equilibrium concept that subjects used is not Nash type but backward elimination of weakly dominated strategies (BEWS). 1 Saijo, Okano, Yoshihara and Saijo (2013) designed the Simplified Approval Mechanism (SAM) in P, and found that the SAM implements cooperation in BEWS. The cooperation rate in the experiment is more than 90% from period one with repetition. Huang, Masuda, Okano and Saijo (2013) expanded the SAM so that the mechanism can handle more than two players. The SAM asks players to choose either cooperation (C) or defection () in the first stage simultaneously. After knowing the choices of all players, all players who chose C have a chance to change from C to simultaneously in the second stage if there is at least a player who chose in the first stage. The SAM implements cooperation in BEWS in theory, but the cooperation rate in early four rounds in the experiment with three players was 64.9%. Any mechanisms should work well from 1 See also Masuda, Okano and Saijo (2013) for a public good provision.

2 period one to have external validity since we cannot apply them in our real life if it takes many periods to achieve the goal of them. In order to cope with this problem, we introduce second thought, not direct punishment and/or reward that have been employed in mechanism design or experiments. 2,3 A player who chose in the first stage has a chance to change the choice from to C in the second thought stage. The SAM with second thought (SAMST) implements cooperation in BEWS. Although the SAM asks a player who chose C to change the decision, the SAMST asks a player who chose to change the decision before asking a player who chose C to change the decision. This is a natural order in our daily life and has been survived in history. Players who are in the second thought stage can observe the choices of the others in the first stage in our design. The observation of others choices might affect the choice in the second thought stage through the mirror mechanism suggested by Gallese and Goldman (1998): it is conceivable that externally-generated MN (mirror neuron) activity serves the purpose of 'retrodicting' the target's mental state, moving backwards from the observed action (p.497). 4 For example, suppose that some subjects chose C and you chose. Then you must consider changing from to C in the second thought stage. The number of C choices other than your choice might affect your decision that we name the eyes of others effect. 5 That is, this paper is a first attempt to consider retrodiction in mechanism design. The introduction of second thought in the SAM does not change the BEWS outcome, but it changes the payoff structure of the game in favor of the success of cooperation. Although the cooperative outcome is attained only when all players choose C in the SAM, the cooperative outcome is also attained even when some of players choose in the first stage since players can change from to C in the second thought stage. In 2 Although punishment in the voluntary contribution mechanism has been used in the literature, it does not have internal validity since it does not implement Pareto efficiency. Furthermore, for example, Guala (2012) found that costly punishment is rare in human history and casted a doubt on its external validity. 3 In the case of not a game but an individual decision making situation, Kessler and Roth (2013) found in the organ donation experiment, a mandated choice frame, which makes individuals select either I want to register as an organ and tissue donor or I do not want to register as an organ and tissue donor, does not increase registration rates and performs directionally worse than an opt in frame in which people check a box to register and leave it blank not to register. Roughly speaking, the mandated choice frame corresponds to the without second thought frame and the opt in frame corresponds to the with second thought frame. 4 See also Gallese and Sinigaglia (2011). 5 A series of experiments by social psychologists such as Haley and Fessler (2005), Bateson, Nettle and Roberts (2006), Ernest-Jones, Nettle, and Bateson (2011) and Nettle, Harper, Kidson, Stone, Penton-Woak and Bateson (2013) shows that a mere picture of eyes in front of subjects makes them cooperative.

3 addition to this nice property, the SAMST is robust against deviation. Even though the probability of deviation from rational choice goes up slightly from zero, the success probability attaining cooperation does not decrease. On the other hand, the success probability in the SAM decreases sharply as the number of players goes up. In the experiments, we examine the effects of second thought for the case of two and three players with 15 periods using random matching. For the case of three players, the cooperation rates are 93.86%, 87.62%, and 19.05% in the SAMST, SAM, and P experiments, respectively. The difference between the SAMST and SAM experiments comes from the earlier periods. The average cooperation rates from period one to four are 86.51% in the SAMST experiment, and 64.88% in the SAM experiment. For the case of two players, on the other hand, the cooperation rates (97.22% in the SAMST experiment and 97.82% in the SAM experiment in all periods) are not significantly different. In the SAMST, for the cases of both two and three players, many subjects who chose in the first stage change the choice, leading that the cooperation rates after the second thought stage are higher than that in the first stage. Consider the eyes of others effect. When one subject chose C and two subjects chose in the first stage, 55.6% of players changed from to C in the second thought stage. When one subject chose C and the other chose in the first stage in the case of two players, it is 60%. When two subjects chose C and a subject chose in the first stage, it is 78.4%. Since the cooperation rate of players increases as the number of C players increases, it seems that the eyes of others effect reasonably works. The organization of the paper is as follows. Section 2 describes theoretical properties of the SAMST. Section 3 presents how the SAMST is robust against deviation compared with the SAM. Section 4 explains the eyes of others effect. Section 5 describes experimental design and procedures and section 6 shows experimental results. Section 7 provides future research avenues. 2. Simplified Approval Mechanism with Second Thought Let n 2 be the number of players, and each player has endowment w > 0. Each player must choose either contributing entire w for the production of a public good y or not. The production function of y is linear, i.e., y = mw where 1> > 1/n and m is the number of players who choose cooperation (C), i.e., contributing entire w. Hence, the payoff of a player who chooses no contribution (defection or ) is mw w ( m 1) w, and contributor s payoff is mw. We call a player who chooses C () a C () player respectively. We will consider a mechanism that has a new stage after the dilemma game due to Huang et al. (2013). If all are C players or they are players, the game ends. The payoff of

4 a player of the former case is nw and the payoff of the latter case is w. If the number of C players is at least one and at most n-1, then only C players can proceed to the second stage and have an opportunity to change their decisions from C to. On the other hand, players cannot proceed to the second stage. This mechanism is called the Simplified Approval Mechanism or the SAM in short. We assume that every player chooses a strategy at each subgame that are not weakly dominated, called Backward Elimination of Weakly ominated Strategies (BEWS). [Figure 1 is around here] Figure 1 illustrates the case with n = 2, = 0.7 and w = 10. Players 1 and 2 face prisoner s dilemma game in the first stage. Knowing that player 2 chose in the first stage at subgame a, player 1 proceeds to the second stage and faces a choice between C and. Player 1 chooses at subgame a since 10 dominates 7, or 10 > 7. Similarly, player 2 chooses at subgame b. Then as the reduced normal form game shows, player 1 chooses C after eliminating weakly dominated strategy. That is, (14,10) weakly dominates (10,10), and hence should be eliminated. Since the remaining strategy is C, player 1 chooses C in the first stage. Similarly, player 2 also chooses C in the first stage, and hence (C,C) is the outcome. We will use the following notation. A strategy profile with parentheses such as (C,C) represents the choice in the reduced normal form game, and a sequence of choices such as CC shows a strategy path or a subgame. Huang et al. (2013) showed the following properties of the SAM. Proposition 1. (i) The simplified approval mechanism implements cooperation in BEWS; and (ii) The simplified approval mechanism cannot implements cooperation in subgame perfect equilibrium. Cooperation in Proposition 1 indicates that all players choose C in the reduced normal form game. As the reduced normal form game in Figure 1 shows, (,) is also a subgame perfect equilibrium (SPE) outcome and hence the SAM cannot implement cooperation in SPE. Huang et al. (2013) conducted experiments of the SAM with three subjects. 21 subjects formed seven groups and then played the SAM and repeated it 15 periods. The groups were formed randomly in each period. The cooperation rates for the first four to

5 seven periods were around 64.9% to 77.7%, and then they were more than 90%. 6 In order to cope with the low cooperation rates around the beginning of rounds, we introduce one more stage called the second thought stage in the following manner. Every player chooses either C or in the first stage simultaneously. If all players choose C or they choose, the game ends. If the number of players is at least one and at most n-1, then players have a chance to change from to C sequentially knowing all choices of the first stage. The order of choices of players are determined exogenously, for example, based upon the order of the numbers assigned to players. Observing the choices of all players in the second thought stage, C players can change the choice from C to simultaneously except for the case where all players change the choice in the second thought stage. When all players change the choice, the game ends and the outcome is that all players choose C. We call this the simplified approval mechanism with second thought (SAMST). Figure 2 shows an example with n = 2, = 0.7 and w = 10. Consider subgame a. Observing player 1 s choice C, player 2 who has a chance to change the choice must consider player 1 s choice at subgame c. Since 10 > 7, i.e., C is dominated by, player 1 will choose at subgame c. Understanding this fact, player 2 at subgame a chooses C since 14 > 10. Therefore, the outcome at subgame a is (14,14) which is different from the outcome of the SAM. Applying the same argument, we have (14,14) at subgame b. That is, the outcomes except for the one at (,) are (14,14) although (14,14) is achieved at (C,C) only in the SAM. [Figure 2 is around here] When the number of players is two, then each player chooses C in the second thought stage, the payoff outcome at (C,) and (,C) is (2 w,2 w). Since the payoff outcomes at (C,C) and (,) are (2 w,2 w) and (w,w) respectively and 2 w > w, the unique SPE strategy profile is (C,C). That is, the simplified approval mechanism with second thought implements cooperation in SPE if n = 2; Figure 3 illustrates the SAMST with n = 3, = 0.7 (0.4 or 0.5) and w = 10. The bold face numbers show payoffs with = 0.7 and the numbers in parentheses show payoffs with = 0.4 or (0.5). Since the entire tree is relatively huge, we show subgames with CCC, CC, C and only, and these are enough to understand the entire tree. Consider first the case with = 0.7. Take a look at subgame a where players 1 and 2 chose C, but 6 Huang et al. (2013) used the ex post cooperation rate. For example, even though a player chose C in the first stage, this was not counted in the cooperation rate if the player changed the decision from C to in the second stage. We also use it in this paper.

6 player 3 chose. Player 3 who faces the second thought stage must consider what would happen at subgame c. Players who chose C in the first stage face prisoner s dilemma game at subgame c, and hence both choose. In this sense, players who chose C in the first stage can burden players who chose in the second thought stage although this hurts every player. Understanding this fact, player 3 compares 21 with C and 10 with. Since C dominates, player 3 chooses C at subgame a. That is, player 3 who chose C at node a can obtain the bonus from players 1 and 2 who chose C in the first stage. Therefore, the outcome of subgame CC is (21,21,21). [Figures 3 and 4 are around here] Consider next subgame b where player 1 chose C, but players 2 and 3 chose. Players 2 and 3 face the second thought stage sequentially. Pay attention to the last nodes where player 1 faces the choice between C and. Although the number of players is one, players who arrive at the nodes face dilemma games and hence they always choose at each node. Consider subgame e where player 2 did not change the choice at subgame b. Since player 1 chooses in the following subgames, player 3 chooses at subgame e and the payoff is 10. On the other hand, player 3 who can take advantage of the bunus effect at subgame d chooses C since player 1 in the following subgame will choose if player 3 choses. That is, 21 > 17. Knowing this process, player 2 chooses C since 21 > 10. Therefore, all payoff outcomes other than (,,) is (21,21,21), and hence the final outcome is (C,C,C) under BEWS as the reduced normal form game in Figure 4-(a) shows. On the other hand, as Figure 4-(b) shows, (21,21,21) appears only at subgame CCC under the SAM. The sequntiality of players is important to implement cooperation. 7 If nodes d and e were in the same information set, the payoff choosing C at the information set would be (21,7) and the payoff choosing would be (17,10), and hence both would survive using the elimination of weakly dominated strategies. Let us take a look at the case with = 0.4. Consider node f where player 1 chooses. Then player 3 at node d chooses since 14 > 12. Knowing this fact, player 2 chooses since 10 > 4. That is, the payoff at subgame b becomes (10,10,10). From the viewpoint of player 3, since is too small, the player cannot take advantage of the bonus effect at node d. As the reduced normal form game in Figure 4-(c) shows, the payoff outcome with subgames where two players choose C and one player chooses is (12,12,12) and the payoff outcome with subgames where one player chooses C and two players choose is (10,10,10), 7 We thank Xiaochuan Huang who pointed out the fact.

7 but the final outcome is still (C,C,C) under BEWS. On the other hand, as Figure 4-(d) shows, (12,12,12) appears only at subgame CCC under the SAM. Consider the case when = 0.5. The payoff outcomes at paths CC and CCC in Figure 3 are (15,5,15) and (15,15,15) respectively. If this is the case, player 3 at node d is indifferent between C and. That is, both C and survive using the elimination of weakly dominated strategies at node d. This influences the decision of player 2 at node b. Figure 5 shows the reduced normal form game at node b excluding player 1. [Figure 5 is around here] Notice that the payoff at (,C) in Figure 5 should be (10,10) when player 2 chooses. That is, player 3 s choice does not matter, and hence both C and survive using the elimination of weakly dominated strategies for player 2. To sum up, the payoff outcome at subgame b or C is either (15,15,15), (15,5,15) or (10,10,10). Similarly, the payoff outcome at subgame C or C is either (15,15,15), (5,15,15), or (10,10,10). That is, there are 3 3 =27 reduced normal form games when = 0.5. Take a look at Figure 4-(e) and consider player 1. There is a case where C and survive using the elimination of weakly dominated strategies for player 1: (10,10,10) at C, (15,15,15) at C and C. Then both players 2 and 3 choose C, and hence the BEWS strategy profiles of the reduced normal form game is (C,C,C) or (,C,C). Although player 1 chooses at (,C,C), the player will change it from to C at the second thought stage, and hence the payoff outcome is (15,15,15). 8 The real problem is that player 1 cannot tell which reduced normal form game player 1 faces and hence both C and survive when player 1 must make a decision at the first stage. Notice that the payoff at C and the payoff at of player 1 is the same for all possible choices of players 2 and 3 in Figure 4-(e). In other words, player 1 cannot distinguish between C and. We say that strategies A and B of a player are indistinguishable if the payoffs at A and B are the same for all possible choices of the other players. Fix a mechanism, and consider all possible reduced normal form games at each node. Strategy A weakly rules strategy B if A weakly dominates B at some reduced normal form game and strategies A and B are indistinguishable at the rest of the reduced normal form games. Then we can define a refinement of BEWS using weak rule instead of weak 8 If we distinguish the case where player 3 chooses C and the case where he chooses at node d and solve the game backwardly for each case, we obtain 8 reduced normal form games. We also have a case where C and survive using the elimination of weakly dominated strategies for player 1: (10,10,10) at C, (15,15,15) at C and C. In this case and any other cases, however, the payoff outcome is (15,15,15).

8 dominance in the definition of BEWS and denote it Backward Elimination of Weakly Ruled Strategies (BEWRS). Take a look at (c) in Figure 4 and consider (,,). This is also a SPE strategy profile, and hence the SAMST cannot implement cooperation in SPE if n = 3. If n > 3, it is easy to find similar examples choosing satisfying 1/(n-1) >. The following proposition shows that the SAMST implements cooperation in BEWS or BEWRS. Proposition 2. (i) If {1/(n-1),1/(n-2),,1/2}, the simplified approval mechanism with second thought implements cooperation in BEWS; and (ii) The simplified approval mechanism with second thought implements cooperation in BEWRS. Proof. (i) Let m and l be the numbers of C and players in the first stage respectively. If m = n, the outcome is ( nw, nw,, nw). If l = n, the outcome is (w,w,,w). Suppose 1 l < n. Consider the choice of players who chose C in the first stage after observing the choices of players in the second thought stage. Let 0 l < l be the number of players who change the choice from to C in the second thought stage. Since (m +l +1)w < (m +l )w + w for all 1 m m-1 where m is the number of C players in the first stage who remains to choose C at the subgame after the second thought stage, is better than C for any C player in the final stage after observing the choices in the second thought stage. That is, all players who chose C in the first stage choose after the second thought stage. Consider any strategy path in which at least one player chose again in the second thought stage. If this is the case, every C player after the second thought stage chooses, and hence the choice does not affect the second thought stage of players. Choose the youngest player (for example, by names or numbers assigned to players) who chose in the second thought stage. Then the subgame after the choice of the youngest player is a sequential social dilemma game, and hence every player after the choice chooses. We will identify the payoff outcome of every subgame constructed by the end nodes of the first stage. Choose any subgame except for the cases where all chose C or all chose in the first stage. Suppose that every player except for the last player changed the choice from to C in the second thought stage. Consider the choice of the last player. If the last player chooses C, the payoff is nw, and if the player chooses, the payoff is (l- 1)w + w. Since n-(l-1) = m+1,

9 if > 1/(m+1), then the last player chooses C; if = 1/(m+1), then the last player is indifferent between C and ; and if < 1/(m+1), then the last player chooses. Suppose > 1/(m+1). If the second player to last chooses, then the payoff is (l- 2)w + w since the last player chooses in the second thought stage and every C player in the first stage will choose in the stage after the second thought stage. If the player chooses C, then it is nw since the last player chooses C. Since nw-{ (l-2)w + w} = { (nl+2)-1}w = { (m+2)-1)}w > 0, the player chooses C. Since nw-{ (l-2)w + w} > 0, nw-{ (lk)w + w} > 0 for all 2 k l. That is, the k-th player to last chooses C, and hence all players choose C in the second thought stage and the payoff outcome is ( nw,, nw). Suppose < 1/(m+1). Then the last player chooses, and hence the payoff of the second to last is (l-1)w if the player chooses C. If the player chooses, then the payoff is (l-2)w + w. Since (l-2)w + w (l-1)w = (1- )w > 0, the player chooses. Since (l-k)w + w (l-k+1)w = (1- )w > 0 for all 2 k l, the k-th player to last chooses, and hence all players in the first stage do not change their decisions in the second thought stage and the payoff outcome is (w,,w). Take any satisfying 1/n < <1 and {1/(n-1),1/(n-2),,1/2}. Consider the case with > 1/2. Then > 1/2 1/(m+1) for all m 1 and hence the payoff outcome of every subgame other than (,,,) is ( nw,, nw): without loss of generality, consider player 1. The payoff at subgame (C,,,,) is nw and the payoff at subgame (,,,) is w. Since nw > w, C is better than. Since > 1/(m+1) for all m 1, the outcome of two subgames (C, ) and (, ) is (C,C,,C) where shows that at least one player s choice is C. That is, player 1 is indifferent between the outcomes of subgames (C, ) and (, ). Therefore, C weakly dominates for all players and hence (C,C,,C) is the BEWS outcome. Consider next the case with 1/2 1/(k+1) > > 1/(k+2) 1/n. Consider player 1. Let indicate that the number of C is k. Then the payoff at subgame (C, ) is nw since > 1/{(k+1)+1} and the payoff at subgame (, ) is w since 1/(k+1) >. That is, C is better than. Since the outcome of two subgames (C, ) and (, ) is the same where indicates that the number of C is not k, C weakly dominates for all players and hence (C,C,,C) is the BEWS outcome. Thus, if {1/(n-1),1/(n-2),,1/2}, the SAMST implements cooperation in BEWS. (ii) Suppose = 1/(m+1). Then the last player is indifferent between C and since nw = (l -1)w+w. Suppose that the second to last chooses C. Then the payoff of the second to last is nw if the last player chooses C and is (l-1)w if the last player chooses. If the

10 second to last player chooses, then the payoff is (l-2)w + w. Since nw { (l-2)w + w} = w > 0, nw > (l-2)w + w > (l-1)w. That is, both C and survive with the elimination of weakly dominated strategies. Since nw > (l-k-1)w + w > (l-k)w for all k=1,..,l-1,both C and survive with the elimination of weakly dominated strategies for all players. Let m( ) (1/ ) 1 where a is the smallest integer not less than a. Since 1/n < < 1, 1 m( ) n 2. Suppose that (1/ ) - 1 is an integer. Then m (1/ ) 1. The following show that there exists a player who is indifferent between C and when the number of C players is m or the number of C players is m 1. Consider two cases: Case 1: Suppose that the number of C players is m. Choose any player who is not a member of the C players. If the player chooses C, the payoff outcome is nw. If the player chooses, the maximum possible payoff is that all players other than the player changes from to C and the player is the last player since all C players change from C to after the second thought stage. Then the payoff is ( l 1) w w and hence nw { ( l 1) w w} = { n ( l 1) -1}w = { ( m 1) 1} w 0 where l n m and n m 2. That is, the payoff of C is the same as the payoff of for the player. l 2 since Case 2: Suppose that the number of C players is m 1. Choose any player who is not a member of the C players. If the player chooses C in the first stage, we will show that the payoff outcome should be at least w. Since l 2, there must be at least one player. If all players changes from to C in the second thought stage, the C player obtains nw. If at least one player chooses in the second thought stage, the C player obtains at least w changing from C to after the second thought stage. If the player chooses, the payoff is w. That is, the payoff of C can be the same as the payoff of for the player. Thus, there is a possibility where C and are indistinguishable for some player. Consider a possibility where C and are indistinguishable for some player. Let player 1 be such a player and suppose that the first m players choose C. Then since C and are indistinguishable, the payoff outcome of subgame ( C, C,..., C ;,..., ) m l the payoff outcome of subgame (, C,..., C ;,..., ). m 1 l

11 Since the payoff outcome of the latter should be (w,w,..,w), each of the last l players in the former can obtain nw changing from to C. That is, C weakly dominates for the last players. On the other hand, compare the payoff outcome of subgame (, C,..., C ;,..., ) m l 1 and the payoff outcome of subgame ( C, C,..., C ;,..., ). The latter payoff outcome should m 1 l 1 be ( nw,, nw), and hence the payoff of player 1 should be nw. Since player 1 at the former should obtain nw that is more than w, there exists at least one player who changes from to C in the second thought stage, and hence all C players who change from C to after the second thought stage should obtain strictly more than w. Then each of the same m players obtain w changing from C to. That is, C weakly dominates for the m players. That is, C and are indistinguishable for player 1 and C weakly dominates for the rest. Suppose that C and are indistinguishable for player 1. Then, there exists another reduced normal form game at the first stage where C weakly dominates for player 1. Since C and are indistinguishable for player 1, for example, the payoff of player 1 at subgame (C,C,..,C;,,) is either nw or w. Since the payoff outcome at this subgame can be either ( nw,, nw) or (w,,w), there is another reduced normal form game where C weakly dominates for player 1. Since the choice of a player who faces indistingushability is arbitrary, C rules weakly for all players. That is, the SAMST implements cooperation in BEWRS. l Consider the meaning of inequality > 1/(m+1), i.e., m > (1/ ) -1. If = 0.7, (1/ ) - 1 = 3/7. That is, the minimum number m of the C players in the first stage where the bonus effect is activated is at least one. If = 0.4, m( ) = 2 which shows that the cooperative outcome at subgame b in Figure 3 cannot be realized since there is only one C player in the first stage. On the other hand, l ( ) n m( ) is the maximum number of players in the first stage so that the cooperative outcome is realized. The proof of the previous proposition shows the following corollary. Corollary 1. If there is an indistinguishable player in a reduced normal form game at the beginning node, no other players are indistinguishable. Suppose that (1/ ) - 1 is an integer. Then there are l 1 possible payoff outcome profiles at a subgame of the second thought stage. The total number of subgames at the second

12 thought stage where the number of players is all possible reduced normal form games at the beginning node is l is n C m and hence the total number of ( l 1) ncm where the number of l combinations from n players. Among these subgames, each player faces just one reduced form game in which C and are indifferent. As Figure 4-(e) shows, the total number of all possible reduced normal form games at the node is (2+1) 3 =27 when n = 3 and m 1. If n = 5 and m 2, player to be indistinguishable is 1/4 10. it is If this is the case, must be 1/3, and the chance of a In order to avoid the indeterminacy when = 1/(m+1), let us introduce an additional behavioral assumption such as risk averse. Player 2 in Figure 5 can obtain 10 if the player chooses at node b. On the other hand, player 2 can obtain either 15 or 5 if player 2 chooses C at node b. If player 2 understands that player 3 is indifferent between C and at node d, player 2 at node b chooses if the player is risk averse. If this is the case, the final outcome should be (,,). That is, the structure of payoff matrix is similar to Figure 4-(c), and hence the final outcome is (C,C,C) under BEWS. Specifically, we say that a player is risk averse if the player faces a choice between (s,t) with probability (1/2,1/2) and u, the player chooses u if (1/2)s+(1/2)t u. Here we implicitly assume that the risk averse player regards the probability of (s,t) as (1/2,1/2) when s and t are indifferent for a player who faces a choice between them. 9 n C k be 3. The SAMST is Robust against eviation Thus far, we suppose that every player chooses an alternative that gives the highest payoff, but we will consider the cases where some player might not choose it due to some wishes such as an occasion where a player chooses in the first stage and keeps the decision hoping that the rest of players who choose C in the first stage will not change the decision after the second thought stage. In addition to wishes, players might make simple mistake or error. Putting together wishes, mistake and error, we call them deviation. For simplicity, we assume that every player makes deviation with probability q with 0 < q < 1 at each node. Take a look at Figure 1. The success probability achieving (14,14) that follows path CC is (1-q) 2. Let CSAM(n,q) be the success probability function of the SAM where n is the 9 The formal proof is as follows: suppose that all players except for the last player in the second thought stage chose C and that the last player is indifferent between C and. Then the second to last player s expected payoff when the player chooses C is (1/2) nw+(1/2) (n-m- 1)w. If the last player chooses, then the payoff of the second to last is (n-m-2)w+w. Since =1/(m+1) and m 1, (n-m-2)w+w -{(1/2) nw+(1/2) (n-m-1)w}=(1/2)w{3-2(m+2) /(m+1)} 0. Since the second to last player is risk averse, the player chooses. Since the rest of players choose, the outcome is (,,,).

13 number of players. Then CSAM(n,q) = (1-q) n. On the other hand, as Figure 2 shows, there are three paths CC, CC, CC achieving (14,14) when we use the SAMST. The probability of paths CC and CC is (1-q)q(1-q) and q(1-q)(1-q) respectively. Hence CSAMST(2,q) = (1-q) 2 + 2q(1-q) 2. The left hand figure in Figure 6 shows the case with n = 2. The horizontal axis shows the probability of deviation and the vertical axis shows the probability achieving cooperation of all players. Consider the case with n = 3. Although CSAMST with n = 2 does not depend on CSAMST with at least three players depend on l that is determined by. That is, CSAMST is a function of n, q, and l and we write it as CSAMST ( n, q, l ). Consider the case with = 0.4. l, Then l (0.4) 1 and there are two types of success paths. The first one is the success paths up to l. They are CCC, CCC, CCC and CCC and the probabilities are (1-q) 3, (1- q) 2 q(1-q), (1-q)q(1-q) 2 and q(1-q) 3 respectively. The second one is the success paths beyond l. Take a look at node b in Figure 3. Although player 2 should choose when = 0.4, this player might choose C by deviation. If player 3 also chooses C right after player 2 s choice by deviation, path CCC is also a success path and the probability of this path is (1-q)q 4. Since there are two other paths of this kind, CSAMST(3,q,1) = (1-q) 3 (1+3q)+3(1-q)q 4. On the other hand, if = 0.7, the probability of CCC is (1-q) 3 q 2. That is, since l (0.7) 2, the deviation in the second thought stage must choose. Therefore, CSAMST(3,q,2) = (1-q) 3 (1+3q+3q 2 ). The right hand figure in Figure 6 shows the case. In order to avoid the indeterminacy case, let us assume {1/(n-1),1/(n-2),,1/2}. Then summarizing the above argument, we have CSAM(n,q) = (1-q) n 1 2 and CSAMST( n, q, l ) = (1 q n ) l C q k n C n k k (1 q ) q. k 0 n k k l 1 n k In order to understand the robustness of the SAMST, compare the slopes of the success probability functions at q = 0. Consider the case with n = 2. Since CSAM(2,0)/ q = -2 and CSAMST(2,0,1)/ q = 0, the success probability of the SAM decreases as q goes up around zero, but the success probability of the SAMST stays at probability one as q goes up around zero. Next, fix any q. Since CSAMST(2,q, ) is always higher than CSAM(2,q) due to 2q(1-q) 2 except for q = 0 or 1, the success probability of the SAMST is always better than that of the SAM excluding the end points. That is, the SAMST is relatively robust enough to handle deviation of players.

14 [Figure 6 is around here] Proposition 3. (i) CSAM(n,0)/ q = -n, and C ( n,0, l )/ q 0 for all l ; and SAMST (ii) For any 1 l n-1, CSAMST(n,q,l) > CSAM(n,q) on (0,1). Then, n l k n 1 n k 2k k 0 n k k l 1 n k n 1 Proof. (i) Let f ( q ) (1 q ), g ( q ) C q and h ( q ) C (1 q ) q. f ( 0) 1. Since f ( q) n(1 q), f (0) n. 0 1 l k l k n 0 n 1 k 2 n k k 2 n k 2 n 1 n k 2( k 1) q r q r( q) (1 ) k l 1 nck q q, g ( q ) C q C q C q 1 nq C q, h( q) ( ) where CSAMST ( n, q, l ) f ( q ) g ( q ) h ( q ), Since l {1,..., n 1} and g( 0) 1 h (0) 0 and. Since g (0) n. Since CSAMST ( n,0, l ) f (0) g(0) f (0) g (0) h (0) n n 0 0. q (ii) By definition, since CSAMST ( n, q, l ) has positive part in addition to CSAM(n,q) on (0,1), we have the result. CSAMST ( n, q, l ) > CSAM(n,q) on (0,1) shows that the SAMST is always better than the SAM with respect to the success probability for cooperation. Since l ( ) is a nondecreasing function, roughly speaking, the success probability goes up as goes up. 4. The Eyes of Others Effect Player 2 at node b in Figure 3 can observe the choices of the others. This observation might affect the choice at node b and there could be many ways to model correct your conduct by observing that of others, retrodiction or simply the eyes of others. Let us consider an example where the observation just affects the deviation probability at the node. Consider the case when =0.7. Then player 2 s right choice at node b should be C with probability 1-q. If nobody had chosen C, player s confidentiality for choosing C would be 1-q. However, since player 1 chose C and player 3 chose before node b, player 2 s choice C is supported by 50% of the other players and should be reinforced. We suppose that this thought process reduces the mistake probability by (1-1/2), and hence the deviation

15 probability at node b becomes (1/2)q. Player 3 at node d observes that players 1 and 2 chose C since player 2 changed the decision from to C at node b. That is, since two out of two chose C, player 3 can confidently choose C with probability 1 and hence the deviation probability becomes (1-2/2)q = 0. Then the success probability of the SAMSM with = 0.7 becomes (1-q) 3 + 3(1-q) 2 q + 3(1-q)q 2 (1-(1/2)q). If = 0.4, it becomes (1-q) 3 + 3(1-q) 2 q+3(1- q)q 2 (1/2)q. Although this change of behavior is very subtle, the success probability goes up around q = 0 considerably due to the eyes of others effect as Figure 7 shows. [Figure 7 is around here] 5. Experimental esign To examine the effects of second thought in the case of two players, we conducted experiments of the two-person prisoner s dilemma game (P2), the two-person simplified approval mechanism (SAM2), and the two-person simplified approval mechanism with second thought (SAMST2). Concerning the case of three players, we reuse the data of prisoner s dilemma (P3) and simplified approval mechanism (SAM3) from Huang et al. (2013), and we conducted the experiment of three-person simplified approval mechanism with second thought (SAMST3). The experiments took place in ecember 2012, February 2013, and June 2013 at Osaka University. The SAM2, SAMST2, and SAMST3 experiments had three sessions, and the P2 experiment had two sessions. 20 subjects participated in each session for the case of two players, and 21 subjects for the case of three players. Total number of subjects is 223. No subject attended in more than one session. We recruited these subjects through campuswide advertisements. They were told that there would be an opportunity to earn money in a research experiment. The data in Huang et al. (2013) include four sessions of the SAM3 experiments and three sessions of the P3 experiments. 21 subjects participated in each session, and hence total number of subjects is 147. Subjects characteristics and the durations of these experimental sessions are summarized in Tables A.1 and A.2 in the appendix. The experiment proceeded as follows. Each subject had a set of printed instructions and a record sheet. Instructions were read aloud by an experimenter. After that, subjects were given five minutes to ask private questions. Communication among subjects was prohibited, and we declared that the experiment would be stopped if it was observed. This never happened. There was no practice period. Each session consisted 15 periods under the random matching protocol. Subjects were informed that they would match the other subjects randomly in every period. We used the z-tree software

16 (Fischbacher, 2007) for the experiment. Let us first explain the SAMST3 experiment. In the first stage, we assigned player number (I, II or III) randomly which showed the decision order in the second thought stage. The payoff table was also displayed on the screen. A subject selected either C or by inputting the choice into a computer, and wrote down it with their reasoning in the record sheet. 10 Then, each subject clicked the OK button. The next was the second thought stage if one or two players chose. The screen of a player showed the choices of all subjects in the first stage. In the screen display of the second player, the choice of the first player was also displayed. They chose either C or, and wrote down it with their reasoning in the record sheet. Then, the subjects who chose C in the first stage proceeded to the final stage if at least one of the players in the second thought stage chose. The screen in the final stage showed the choices of all subjects in the first and the second thought stages, and the payoff table. Then, subjects chose C or, and wrote down it with their reasoning in the record sheet. Then, the outcome screen displayed the choices of all subjects in the first, the second thought, and final stages, the final choices, and the payoffs. This ended one period. The experimental procedure in the SAMST2 experiment is the same above. The experimental procedure without the second thought stage becomes that in the SAM3 and SAM2. The experimental procedure without second thought and final stage becomes that in the P3 and P2. After finishing 15 periods, each subject completed a questionnaire and was immediately paid privately in cash. Each obtained money that was proportional to the sum of the points earned for the 15 periods. In session one of the SAM2 experiment, subject 19 chose in all periods. The questionnaire showed that the subject could not understand the game rule throughout all periods. We decided to eliminate the data of this subject and the counterparts. 6. Experimental Results 6.1 Cooperation Rates We first provide the experimental results for the case of two players. Figure 8 shows the average cooperation rates across periods. The cooperation rate is the number of C over the number of all choices. We use the ex post cooperation rate for the SAM2 and SAMST2 experiments: for example, in the SAM2 experiment, if a player chose C and the other player chose in the first stage, and the player changed from C to in the final stage, then we counted that choice as defection. 10 In the experiment, we used A (defection) and B (cooperation) as the strategy label rather than C and.

17 [Figure 8 is around here] The average cooperation rates are 97.22% and 97.82% in the SAMST2 and SAM2 experiments, respectively, which are not significantly different (p = 0.800, the Wilcoxon rank-sum test). 11,12 For the case of two players, we cannot confirm the effect of second thought because there is almost no room for the SAMST2 to improve the behavior of subjects compared with the SAM2. On the other hand, the average cooperation rate is 6.33% in the P2 experiment. The cooperation rates in the SAMST2 and SAM2 are significantly higher than that in the P2 (p < 0.001, the Wilcoxon rank-sum tests), which indicates that both the SAMST2 and SAM2 make subjects more cooperative. Figure 9 shows the cooperation rates for the case of three players. The average cooperation rates are 93.86%, 87.62%, and 19.05% in the SAMST3, SAM3, and P3 experiments, respectively. As in the case of two players, the cooperation rates in the SAMST3 and SAM3 are significantly higher than that in the P3 (p < 0.001, the Wilcoxon rank-sum tests). [Figure 9 is around here] By contrast to the case of two players, the cooperation rates are significantly different between the SAMST3 and SAM3 experiments (p < 0.001, the Wilcoxon rank-sum test). This difference comes from the earlier periods. In period one, the cooperation rate is 76.19% in the SAMST3, while 54.76% in the SAM3. Then, the cooperation rate in the SAM3 substantially remains lower than that in the SAMST3 from period one to four. The cooperation rates are significantly different in period one, three and four at the 5% significant level, and in period two at the 10% significant level (p = in period one, p = in period two, p = in period three, and p = in period four, the chi-square tests). This indicates that second thought effectively works to improve the cooperation rates in earlier periods. Result In the Wilcoxon rank-sum test, we first calculate the average cooperation rate of each subject across periods and then calculate the test statistic using the averages in order to eliminate correlation across periods. 12 The comparisons of the cooperation rates for each period also reveal that they are not significantly different between SAMST2 and SAM2 experiments in all periods (p > 0.1 in all periods, the chisquare tests).

18 (i) The average cooperation rates are 97.22%, 97.82%, and 6.33% in the SAMST2, SAM2, and P2 experiments, respectively, and the cooperation rates in the SAMST2 and SAM2 are not significantly different, and significantly higher than that in the P2. (ii) The average cooperation rates are 93.86%, 87.62%, and 19.05% in the SAMST3, SAM3, and P3 experiments, respectively, and the cooperation rate in the SAMST3 is significantly higher than that in the SAM3, and the cooperation rates in the SAMST3 and SAM3 is significantly higher than that in the P Robustness of Second Thought against the eviation The SAMST is robust against the deviation from BEWS prediction, as argued in section 3. We examine this feature from the experimental data. We first compare the average cooperation rates in the first stage and after the final stage. The average cooperation rates in the first stage are 94.60% in the SAM3, and 90.26% in the SAMST3, which is not statistically different (p = 0.770, the Wilcoxon ranksum test). The cooperation rate after the final stage declines to 87.62% in the SAM3. Note that in the SAM, the cooperation rate should not increase from the first stage to the final stage because C players can change from C to, while players must stay at. In the SAMST, the final cooperation rate might increase because players can change the choice from to C in the second thought stage. Actually, the cooperation rate increases to 93.86% after the final stage. For the case of two players, the cooperation rates in the first stage are 98.85% in the SAM2, and 96.44% in the SAMST2, which is not statistically different (p = 0.802, the Wilcoxon rank-sum test). As in the case of three players, the cooperation rate in the first stage decreases after the final stage in the SAM2 (from 98.85% to 97.82%), while it increases in the SAMST2 (from 96.44% to 97.22%). Result 2. (i) The cooperation rate decreases from 94.60% in the first stage to 87.62% in the final stage in the SAM3, and from 98.85% to 97.82% in the SAM2 experiment; and (ii) The cooperation rate increases from 90.26% in the first stage to 93.86% in the final stage in the SAMST3, and from 96.44% to 97.22% in the SAMST2. Let us calculate the probability of deviation using the success rates. 13 In the case of three players, the probabilities of deviation are 15.6% and 4.8% since the success rates are 92.7% and 86.2% in the SAMST3 and SAM3, respectively using the right figure in 13 The success rate is defined as the rate at which all players in the group end up with C.

19 Figure 6. Therefore, the probability of the deviation is higher in the SAMST3 than in the SAM3, although the success rate is higher in the SAMST3 than in the SAM3. Figure 10 reports the probability of the deviation across periods for the case of three players. In both the SAMST3 and SAM3, the probability of deviation generally declines from period 1 to 5. Then it stays at low level in the SAM3, while it is volatile in the SAMST3. The hypothesis test based on Spearman s rank correlation coefficient shows that the downward trend of the probability is statistically significant in the SAM3 ( = , p = 0.005), while not in the SAMST3 ( = , p = 0.175). Together with the high cooperation rates in the SAMST3, these results indicate the robustness of second thought against the deviation. [Figure 10 is around here] For the case of two players, the probabilities of deviation are 10.2% and 1.2% since the success rates are 97.1% and 97.7% in the SAMST2 and SAM2, respectively, using the left figure in Figure 6. As in the case of three players, the probability of the deviation is higher in the SAMST2 than in the SAM2, although the success rate is higher in the SAMST2 than in the SAM2. Figure 11 reports the probability of the deviation across periods for the case of two players. In the SAM2, the probability of the deviation is at low level in all periods, while it is volatile in the SAMST2. Spearman s rank correlation coefficients indicates that either increasing or decreasing trend of the probability is not statistically significant in both the SAM2 and SAMST2 ( = , p = in the SAM2 and = 0.316, p = in the SAMST2). Furthermore, the probability of the deviation is lower in the case of two players than in the case of three players. [Figure 11 is around here] Subjects seem to deviate from the BEWS prediction for various reasons. Let us take a look at the record sheet. The most prominent description for the deviation is that some subjects seem to aim at getting the highest payoff (2400 in the case of three players and 1700 in the case of two players) which is obtained when they choose and the other(s) choose(s) C. The other descriptions are that some subjects seem to want to know what outcome will happen by choosing, or that some other subjects seems to avert the lowest payoff (which is obtained when they choose C and the other(s) choose(s) ), and so on.

20 Result 3. (i) The probabilities of the deviation are 15.6% and 4.8% in the SAMST3 and SAM3, respectively. (ii) The probability of the deviation in the SAM3 significantly decreases over periods, but does not in the SAMST3. (iii) The probabilities of the deviation are 10.2% and 1.2% in the SAMST2 and SAM2, respectively. (iv) The probabilities of the deviation in the SAMST2 and SAM2 do not significantly decrease over periods. 6.3 The Eyes of Others Effect Players can observe the choices of the other players after the first stage. For example, player 3, who is a player, at subgame CC in Figure 3 observes that all other players chose C. Consider the number (percentage) of the other player(s) who choose(s) C at each subgame. At subgames C and C in Figure 2, one player (100%) chooses C. At subgames CC and CC in Figure 3, two players (100%) choose C. At subgame C one player (50%) chooses C. We will investigate whether these numbers (percentages) affect the subjects choice in the second thought stage. Note that subgames CC and CC in Figure 3 are quite different. At subgame CC, the first player who changed from to C in the second thought stage cannot change the choice while at subgame CC, two C players can change the choice in the final stage. Therefore, we should distinguish between CC and CC. Table 1 shows the frequencies and percentages of choices in the second thought stage. The frequencies in four categories are statistically different at the 10% significant level (p = 0.073, the chi-square test). [Table 1 is around here] Let us compare Categories (i) and (ii) in Table 1. This comparison allows us to see the effect when the number of C players increases from one to two while keeping the percentage. The cooperation rates in the second thought stage are 60% and 78.4% in Categories (i) and (ii) respectively. This difference is statistically significant at the 10% significant level (p = 0.056, the chi-square test). This indicates that the increase in the number of other subjects with C facilitates the cooperative behavior. Comparison between Categories (i) and (iv) also provides the same effect, but we observe the effect of the opposite direction. The cooperation rate in the second thought stage is 40% in Category (iv), which is