Framing and cooperation in public good games : an experiment with an interior solution 1
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1 Framng and cooperaton n publc good games : an experment wth an nteror soluton Marc Wllnger, Anthony Zegelmeyer Bureau d Econome Théorque et Applquée, Unversté Lous Pasteur, 38 boulevard d Anvers, Strasbourg, France Abstract We show that expermental subects tend to contrbute more to the publc good f they perceve the actons of the others as a source of postve externalty rather than a source of negatve externalty. In our experment partal contrbuton to the publc good s the unque subgame perfect equlbrum for the repeated game. Résumé Dans cet artcle, nous montrons que le degré de coopératon de suets placés dans un mécansme de contrbuton volontare est dfférent suvant que les chox des suets ndusent des externaltés postves (contexte postf) ou des externaltés négatves (contexte négatf). Les suets partcpant au contexte postf ont un taux de sur-contrbuton supéreur au taux de sur-contrbuton des suets partcpant au contexte négatf. Notre expérence se caractérse par une stratége domnante d nvestssement partel dans le ben publc. Keywords: Expermental economcs; Publc goods JEL classfcaton: C9; H. Introducton Observed cooperaton n publc good experments has not yet receved a fully satsfactory explanaton. Cooperaton n publc good experments contrasts sharply wth behavour observed n olgopoly experments and common pool resources (CPR) experments. Although, these experments have the same strategc envronment as publc good games, the observed results are, n ths case, compatble wth the game-theoretc predctons (see Ostrom et al., 99). One possble reason s that n the publc good envronment subects perceve the actons of the others as a source of postve externalty, whle n the olgopoly or CPR envronment they perceve those actons as a source of negatve externalty. In order to test ths hypothess, we desgned an experment smlar to Andreon (995) whch allows a comparson of the two contexts. In the postve context, any token nvested by a member of the group n the publc actvty, pays off the same amount to each member of the group. In the negatve context, any token nvested by a member of the group n her prvate actvty generates a loss for the other members of the group. Andreon (995) found that the average level of contrbuton s sgnfcantly larger n the postve context compared to the negatve context. However, hs result could be due to a partcular combnaton of desgn optons. Andreon chose a corner soluton desgn where the domnant strategy s to contrbute nothng to the publc good, and a strangers desgn n whch subects are randomly assgned to one of fve groups after each perod. When the domnant strategy s to contrbute nothng cooperaton can be due to decson errors. Moreover, n the negatve context the strangers desgn could have renforced the feelng of adversty nduced by the actons taken by the other subects. Our experment s based on We would lke to thank Clauda Keser for valuable comments and conversatons. Correspondng author. Tel. : , fax: , emal : wllma@cournot.u-strasbg.fr.
2 a publc good game for whch the domnant strategy s to contrbute a postve amount to the publc good (see Keser (996)). We chose a partners desgn, for whch subects are randomly assgned to a group for the whole duraton of the experment. Secton presents the expermental desgn, secton 3 summarzes the results and secton concludes by a short dscusson.. Expermental Desgn Each partcpant was randomly assgned to a group of four people (ncludng hmself) for the 5 perods of the experment. At the begnnng of each perod, each subect was endowed wth 0 tokens, whch he was asked to allocate between a prvate actvty and a publc actvty. The consttuent game has a domnant strategy equlbrum where each player contrbutes 7 tokens to the publc actvty, whch s also the level of contrbuton predcted by the unque subgame perfect equlbrum of the repeated game. Because we have an nteror soluton, over-contrbuton can hardly be explaned by decson-errors snce the cost of a devaton s the same n both drectons. We studed two treatments : the postve treatment s a standard publc good game wth an nteror soluton. We replcated the experment of Keser (996) wth 5 perods nstead of 5. In the postve treatment each token nvested n the publc good generates a unform gan for each member of the group. In the negatve treatment, each token nvested n the prvate actvty generates a loss for the other members of the group. The structure of the game s dentcal n both treatments. Only the presentaton dffers... The postve treatment In the postve treatment, the ndvdual payoff functon of agent s gven by: = ( x, t ) = x x + 5 t. () Each player maxmzes Π wth respect to (x, t ) and subect to the budget constrant x + t = 0, where x s the number of tokens nvested n the prvate actvty and t the number of tokens nvested n the publc actvty. The reward of the prvate actvty s x x whle the reward of the publc actvty s gven by 5 t. In the postve treatment, contrbuton to the publc actvty = generates a postve externalty for other subects. Ths becomes clear f we rewrte expresson () as: = ( x, t ) = x x + 5 t. () # The unque Nash equlbrum s x * = 3 and t * = 7. Moreover the repeated game has a unque subgame-perfect equlbrum where each player contrbutes seven tokens per perod to the publc actvty. The socal optmum requres that each member contrbutes 0 tokens to the publc actvty... The negatve treatment The negatve treatment s smply a rewrtng of the reward structure, n whch nvestng n the prvate actvty appears as generatng a negatve externalty for the other members of the group. Formally, ths s done by takng nto account the budget constrant of the other players nto player s payoff functon (): #
3 ( x, x ) = x x + 5 (0 - x ) (3) # ( x, x ) = x x x. () # The last term of the RHS of equaton () shows that tokens nvested by the three other players generate a loss for agent. Of course, the framng has no effect on the game tself. A self-nterested player stll has a domnant strategy to contrbute seven tokens to the publc actvty. Also, the subgame-perfect equlbrum of the 5-perod game and the socal optmum are the same than n the postve treatment. Thus, the ncentves n both treatments are dentcal. They only dffer by the framng of the context. Except for the necessary changes requred by our partcular desgn, the nstructons are smlar to those of Andreon..3. Practcal procedures The experment was run on a computer network n Sprng 996. We used the RatImage (995) toolbox n order to compute the experment. Four sessons were organzed, wth 3 groups of subects per sesson. A total of 6 ndependent observatons per treatment was collected. The subects were recruted by phone from a pool of 500 students, who were wllng to partcpate n an experment. Subects earned 30 FF show-up fee along wth ther earnngs n the experment. The average earnng was 89 FF, for sessons whch lasted about one hour and a half. Subects were randomly assgned to a group of four players, to play a 5-fold repetton of the oneshot game, on a computer termnal, whch was physcally solated from other termnals. Communcaton was not allowed. The subects were nstructed about the rules of the game and the use of the computer program through wrtten nstructons (avalable upon request), whch were read aloud. A short questonnare and two practce rounds followed. Subects earned ponts that were converted at the end of the sesson. The number of ponts accumulated snce the begnnng of the experment was on permanent dsplay and subects could vew the payoff hstory. At the end of a sesson each subect was pad prvately the total amount he/she had earned durng the sesson. # # 3. Results Table shows, for each ndependent player group, the average token contrbuton to the publc actvty for the 5 perods and for both treatments. In the postve treatment, the average contrbuton per subect and per perod s.39. Ths corresponds to an average per perod overcontrbuton rate of.6 % per subect. In the negatve treatment, the average contrbuton to the publc actvty per subect and per perod s 8.6. Ths corresponds to an average per perod overcontrbuton rate of 9.69 % per subect. Table Average contrbuton for the ndependent player groups, postve groups (+ to 6+) and negatve groups (- to 6-) Group Average contrbuton 3,7,63,53 9,78 3,0,0 9,0 7,80 9,0 6,98 0,00 6,7 t 7 The ndvdual over-contrbuton rate s defned by. 3 3
4 In the postve treatment, the average contrbuton s above the subgame-perfect equlbrum level of contrbuton (7) for each group. A χ test reects the null hypothess that the average levels of contrbuton are equal to 7 at the % level. Ths result confrms earler results that showed that subects sgnfcantly over-contrbute to the publc actvty wth respect to the domnant strategy. In the negatve treatment, the average contrbuton s not always above seven for each group. A χ test does not reect the null hypothess that the average levels of contrbuton are equal to 7 at the 5 % level. Thus, the negatve treatment of the publc good game wth nteror soluton agrees wth the equlbrum predcton. Fg. shows the tme path of the average contrbutons to the publc actvty for both treatments. In each perod the average contrbutons are above the domnant soluton of seven tokens n the postve framng n contrast to the average contrbutons n the negatve framng. tokens Negatve framng Postve framng Domnant strategy perod Fg.. Tme path of the average contrbutons to the publc actvty for both treatments. Applyng a two-taled Mann-Whtney U test, we can reect the null hypothess that average contrbuton levels are the same n both treatments at the % level. We conclude that subects contrbute sgnfcantly more n the postve framng than n the negatve framng. Ths result s n lne wth the observaton made by Andreon (995). But n contrast to hs results, we observe no sgnfcant dfference n contrbuton levels n the frst perod n the two contexts. The gap arses only wth the repetton of the one shot game, suggestng that t s not the percepton of the envronment as such that nduces lower contrbuton n the negatve context, but the percepton of the others effectve contrbuton.. Dscusson and concluson We desgned an experment to test Andreon s observaton that subects contrbute more to the publc good when they perceve the actons of the others as a postve externalty rather than a negatve externalty. In contrast to Andreon, our expermental desgn s based on an nteror soluton and on a partners treatment. We observe that the average level of contrbuton n the negatve treatment s sgnfcantly lower than n the postve treatment and agrees wth the equlbrum predcton. Lookng at ndvdual contrbuton levels, the dfference mght attrbuted to
5 the fact that more subects over-contrbuted n the postve treatment. 80,56 % of the subects contrbuted more than 7 tokens n the postve treatment aganst 3,33 % only n the negatve treatment. The dfference s sgnfcant at the % level. Moreover, for any nteger α, such that α 3, there are more subects contrbutng at least 7+α tokens to the publc good n the postve treatment than n the negatve treatment (the dfference s always sgnfcant at the % level). In the frst perod the average level of contrbuton n both treatments s not sgnfcantly dfferent. Snce the dfference becomes sgnfcant only wth the repetton of the game, t must be the case that, on average, subects react n the dfferent way to the contrbuton of the others n the two contexts. However, a better understandng would requre a detaled characterzaton of ndvdual play n both contexts. References Andreon, J., 995, Warm-glow versus cold-prckle: the effects of postve and negatve framng on cooperaton n experments, The Quarterly Journal of Economcs 60, -. Keser, C., 996, Voluntary contrbutons to a publc good when partal contrbuton s a domnant strategy, Economcs Letters 50, Ostrom, E., Gardner, R. and J. Walker, 99, Rules, games, and common-pool resources, Unversty of Mchgan Press, Mchgan. RatImage, 995, Research assstance toolbox for computer-aded human behavor experments, Klaus Abbnk and Abdolkarm Sadreh, Unversty of Bonn. 5
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