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Appendix 2: Charity We assume that people who are asked to give a favor know with certainty the amount of charity that others have contributed. Suppose others believe that if a person contributes charity ofamount C, she will be a reciprocator forever, and if she contributes C* > C, she will be a favor initiator forever, and if she gives less than C, she will be either a moocher or a nonplayer. Are there a C and a C* that will make that belief self-fulfilling? Calculate the maximum C that any moocher will be willing to pay in charitable contributions to be confused with a reciprocator . The expected present value of the moocher's return if she gives less than Cis 0, since nobody will do her a favor. If she gives C, her expected gross return (not including her charitable contribution) is given by equation (A.I), assuming initially that everybody remembers forever how much everybody has contributed to charity. So set C equal to that gross return at a gain level that just separates moochers from others. Now set C* so that it is the smallest distinguishable value greater than C. Reciprocators do not gain from being confused with favor initiators, since they would refuse an initial request for a favor if it were made. However, favor initiators do gain from being identified as favor initiators because it pays for them to initiate such favors. Hence, the slightest contribution above C will serve to separate reciprocators from favor initiators. To determine C from equation (A.I) it is necessary to determine the maximum gi such that i will be a moocher. In the simple charity casewhere only gi varies-the gi such that people are indifferent between mooching and reciprocating is the same as the gi such that they are indifferent between mooching and favor initiating. Hence, gI =g2 =g , where gI is the gi such that Mi =Ri;g2 is the gi such that Ri =Fi; and J3 is the gi such that Mi = Fi· g] = g2 = g3 = f(1 + r) / (1 - P). (AA) 207 208 Appendix 2 The key to equation (A-4) is that P2' the probability that a favor will be reciprocated, is now equal to I for both favor initiators and reciprocators , since favor initiators will confine their largesse to those who have contributed to charity, who are either favor initiators or reciprocators . Bygones are bygones. Reciprocators act as if they were favor initiators when it is their turn to reciprocate. With certainty that their favor will be reciprocated, favor initiators get the same return for a given gi as do reciprocators at the time that reciprocators reciprocate. In consequence, the gi that is required to induce either to assume their respective roles will be the same. Then C is simply the expected returns to mooching at g,: equation (A.I) calculated at g" or (A.S) Now consider the relationship between the gains of players and charity. Some determinants ofgains vary within a distribution ofgains if these are characteristics that are unknown to the players. On the other hand, known characteristics are parameters determining a particular distribution. (We assume that people sample at random within a distribution or within a subset determined solely by signaling. This assumption is appropriate only if they sample within a distribution for which the only information about trustworthiness known to others is the signal.) Within any given distribution of gains, those with greater gains are more likely to give to charity, since they are more likely to be favor initiators . But what happens to charity as the whole distribution of gains changes? The variables affecting C in equation (A.S) are not related to the distribution of gains, not even P, the probability of a favor request's being granted, even though without charity, P is a function of that distribution. In the charity case both the requests for favors and the responses come from the same group: favor initiators. There will be no pure reciprocators, and moochers are screened out. P, then, depends solely on the ratio of unmatched to total favor initiators. In the steady state that ratio will be determined solely by the probability that people stay in the market another period. But though C is unrelated to the distribution of gains, the expected amount of charitable contributions per capita will be closely related. C is the amount given by those who give to charity. The expected per capita amount of charity is C...

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