Problems to be discussed at the 5 th seminar Suggested solutions

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1 ECON4260 Behavoral Economcs Problems to be dscussed at the 5 th semnar Suggested solutons Problem 1 a) Consder an ultmatum game n whch the proposer gets, ntally, 100 NOK. Assume that both the proposer (A) and the responder (B) have nequty averson as specfed n the model of Fehr and Schmdt (1999), wth α = 3 and β = 0.3, = A,B. Explan the optmal strategy of the two players f these preferences are common knowledge. In the 2-player verson of the FS (1999) model, preferences are specfed as U x max x x,0 max x x,0 where j, and β α, 0 β < 1. ( prefers that j s ncome s equal to hers; s utlty declnes n ther ncome dfference, more so f herself s worst off.) Usng the parameter values n 1a), we get that U x x,0 0.3max x x,0 x 3max j j Utlty conssts of materal payoff mnus a loss from nequty. A rejected offer thus always mples a utlty of zero for both players (n ths case, the outcome s symmetrcal): no materal payoff and no nequty. U 0 3max 0 0,0 0.3max 0 0,0 0 Consder frst the Responder s choce. Any offers s 0.5 wll be accepted: If s=0.5, and the offer s accepted, there s no loss from nequty, but materal payoff s strctly postve. Hence the offer s accepted. (Ths could of course be shown formally - take a look at the utlty functon to check that the clam makes sense.) If s > 0.5, the Responder s utlty s U x 0.3( x x ) x 0.3x 0.3x 0.7x 0.3x whch s strctly postve regardless of how the 100 NOK are shared,.e. larger than the utlty of Reject, so offer s accepted. (Alternatvely, you can solve w.r.t s, just lke below.) To decde whether offers below 0.5 wll be accepted, we must compare the Responder s utlty of rejectng whch s 0 to the case n whch A gets X(1-s) and B gets sx. If the responder accepts, hs utlty wll be (usng that s<0.5, so gets more than j) U 100s 3max 100(1 s) 100 s, 0 0.3max 100s 100(1 s), 0 B B B A B B A B A 100s 3 (1 s) s 100( s 3 3s 3 s) 100(7s 3) j j

2 Ths s less than 0, and thus worse than reject, f 100(7s 3) < 0.e. 7s<3 s<3/7 Hence, the Responder wll accept anythng above 3/7 (and s ndfferent between Reject and Accept when s=3/7). Consder then the Proposer (A). Snce β A = 0.3<0.5, he wll prefer to keep everythng for hmself (see notes to lecture 2). But snce he knows that B wll reject f s<3/7, he wll offer exactly 3/7 (or just slghtly more to ensure strct preference), and B wll accept. Ths s the quck&easy answer, whch s a lttle unsatsfactory because I just refer to a result from the lterature wthout explanng t. Your answer wll be better f you show that a person wth β A = 0.3 wll prefer to keep everythng to hmself, or explan ntutvely why, n general, A wll prefer to keep everythng when β A <0.5. Formally: Consder the case where s<0.5 (a smlar analyss can be done for s 0,5). Then U 100(1 s) 0.3(100(1 s) 100 s) A 100(1 s 0.3(1 s s)) 100(1 s s) 100( s) Then one can see that A s utlty s decreasng n s: UA/ s = Thus, A wll prefer s as low as possble. Why, n general, wll a person wth β <0.5 prefer to keep everythng to hmself? Intutve explanaton: When A gves B one kr, A loses one kr, but nequty s changed by 2 kr (A gets 1 kr poorer and B gets one kr rcher). Thus, to want to gve money away, A must place a weght on nequty whch s at least half as bg as the weght he places on ncome. (When s<0.5, A s rchest, so ths reasonng means that β 0.5. If s>0.5, B s rchest and α s the relevant parameter; but we know that α β that s an assumpton of the model, so f beta sn t bg enough, alpha won t be ether.) (Or: When A s better off, the utlty functon can be wrtten UA= xa βa (xa-xb) To see what happens f A gves 1 kr to B, we can dfferentate and use -1=dxA=-dxB: dua= dxa βa (dxa-dxb) = -1 βa (-1-1) = βA A s exactly ndfferent when (-1 + 2βA ) = 0,.e. βa = ½.) b) Assume that the responder (B) has the preferences specfed n Queston 1a), but that the proposer (A) s only concerned about hs own materal payoff. What are now the players optmal strateges, f these preferences are common knowledge? The answer s exactly lke n 1a). The Proposer s averson aganst advantageous nequty n queston 1a) s nsuffcent to make hm want to share. He shares only because B would otherwse reject. Ths corresponds exactly to the behavor of a self-nterested Proposer. c) Assume that the proposer (A) has the preferences specfed n Queston 1a), but that the responder (B) s only concerned about hs own materal payoff. What

3 s now the optmal strategy for the two players, assumng that ther preferences are common knowledge? The Proposer stll does not want to share, as n 1a). The Responder now does not have any credble threat to reject strctly postve amounts, snce the Responder knows hs preferences. Hence, predctons are exactly lke n the self-nterest case (see lecture notes, Lecture 1 of Topc 3): If the lowest possble amount s 50 øre, there are two subgame perfect Nash equlbra: One n whch the proposer proposes 50 øre and the responder accepts any strctly postve offer, and another n whch the proposer proposes nothng and the responder accepts anythng. Problem 2 Consder the followng game: Cooperate Defect Cooperate 3,3-5, 5 Defect 5,-5 0,0 The rows correspond to possble choces of Player A, whle columns correspond to possble choces of Player B. The frst number n each cell denotes player A s materal payoff, the second number n each cell denotes B s materal payoff. Thnk of A and B as partners n a frm. If both nvest 10 n a project, the project wll acheve an ncome of 13 (per person), so both wll get net earnngs of 3. If only one of them nvests, the project earns only 5 (per person), leadng to a payoff of -5 for the person who nvested and 5 for the other. If none of them nvests, both get nothng. a) What are the players domnant strateges f each cares only about hs own materal payoff? Ths s a Prsoners Dlemma game, so Defect (do not nvest) s a domnant strategy for both players: Consder the poston of player A. If player B cooperates (nvests), A gets 3 f he nvests too, and 5 f he defects (does not nvest). If player B defects (does not nvest), A gets 5 f he nvests, and 0 f he does not nvest. Hence, regardless of what B does, A s better off by not nvestng, so Defect (do not nvest) s A s domnant strategy. By symmetry, the same holds for B. (So (Defect, Defect) s the only Nash equlbrum of ths game.) b) Assume that both players have nequty averson as specfed n Queston 1a). Is (Cooperate, Cooperate) then a Nash Equlbrum? Assume frst that player B cooperates (nvests). Then, f A cooperates (nvests) too, both get a materal payoff of 3, so A s utlty s U x 3max x x,0 0.3max x x, j j (snce there s no nequty). The same holds for B (hs utlty s also 3). Smlarly, f both defect, there s no nequty, and each player s utlty equals hs materal payoff, namely 0. If A defects and B cooperates, A s utlty s U (5 ( 5)) A

4 whch s less than hs materal payoff because he dslkes dsadvantageous nequty. (If A cooperates and B defects, A s payoff s 5 3(5 ( 5)) U A but ths s not really needed to arrve at the answer that C,C s a Nash eq.). Assume now that both nvest (cooperate). Then A s utlty s 3. If he changes hs strategy and defects, whle B s strategy s kept fxed, A s utlty s 2, whch s lower than 3. By symmetry, the same reasonng holds for B. Consequently, (Cooperate, Cooperate) s a Nash equlbrum wth these preferences. You can see ths clearly by wrtng down a payoff matrx n utltes rather than money. Although some numbers n ths matrx are, strctly speakng, not needed to arrve at the answer, ths s a nce way to provde an overvew. Cooperate Defect Cooperate 3,3-35, 2 Defect 2,-35 0,0 It s now straghtforward to verfy that C s each player s best response to the other playng C. (D,D) s also a Nash equlbrum. c) Assume that player A has nequty averson as specfed n Queston 1a), but that player B cares only about hs own materal payoff. Is (Cooperate, Cooperate) then a Nash Equlbrum? A s payoffs are as derved n queston b) above. However, B cares only about hs materal payoff, whch s 3 n the case of mutual cooperaton and 5 f he defects and A cooperates. Hence, (C,C) cannot be a Nash equlbrum: Compared to ths stuaton, player B can gan by changng hs strategy to Devate. Utltes are now Cooperate Defect Cooperate 3,3-35, 5 Defect 2,-5 0,0 so D,D s now the only Nash equlbrum. Problem 3 Assume now that both players n the game descrbed n Problem 2 above have recprocal preferences. Let player s utlty U be defned as follows: U x k k j j

5 where x = s materal payoff, kj = s kndness towards j, and kndness towards (=1,2; j=1,2; j). k j = s belef about j s 3 a) Does t seem reasonable that a person wth recprocal preferences as specfed above mght prefer to play Cooperate n ths game, f he expects that the other wll play Cooperate? Explan the man ntuton (you should not necessarly have to defne kndness formally to be able to do ths). If the other s knd, he wants to be knd n return; f the other s mean, he wants to be mean n return. It seems ntutvely reasonable to classfy cooperate as a knd acton, trggerng a desre to be knd n return. If ths recprocal preference s strong enough, t may outwegh the concern for materal payoff (n a farly smlar fashon as n the nequty averson example n Semnar 5). (Note: The ntutve reasonng that perceves j as knd f j Cooperates because j s sacrfcng some of hs materal payoff to help, s perfectly ok. But note that ths partcular dea s not captured n the formal defnton of kndness below, snce that defnton does not nclude j s payoff (hs sacrfce) n the measurement of j s kndness, only the payoff that j secures to.) 3 b) Explan what a farness equlbrum s. A farness equlbrum s a stuaton n whch every player maxmzes hs utlty, gven hs belefs and the other players strateges, and where belefs are correct. Thus, n a farness equlbrum, no player has any reason to change hs strategy and/or hs belefs, gven the strateges and belefs of the other players. In standard game theory, t s assumed that players payoffs (utlty) depend only on outcomes, not on belefs per se. In models of recprocty, utlty can depend drectly on belefs, not just on outcomes (e.g.; I may feel bad f I beleve that your ntentons were mean, even f I m wrong), thus volatng ths assumpton from standard game theory. The concept of farness equlbrum s qute smlar to a Nash equlbrum, but s developed for models n whch belefs may matter drectly for payoffs. Let us now defne kndness formally. Assume that s kndness towards j s defned n the followng way: k j = x j (s, b j ) - 1/2[x j max (b j )+ x j mn (b j )] where s = s strategy (Cooperate, or Defect), b j = s belef about j s strategy (Cooperate, or Defect), x j max (b j ) s the largest materal payoff could secure to j, gven s belef about j s strategy b j, and x j mn (b j ) s the smallest materal payoff could secure to j, gven b j. Hence, kndness s gven by the payoff allocates to j compared to the average of those payoffs could potentally have allocated to j (gven hs belefs). Moreover, let c j = s belef about j s belef about s strategy. We can then defne same way as k j, but takng nto account that to evaluate j s kndness towards hmself (), must use hs belefs about j s strategy, b j, and hs belefs about j s belefs about s own strategy (c j ): k j n the

6 x (b j, c j ) - 1/2[x max (c j )+ x mn (c j )] 3 c) What s/are the farness equlbrum/equlbra n the game descrbed above, gven the recprocal preferences and defntons of kndness specfed here? Explan. Only states where belefs are correct can be canddates for farness equlbra. For a state to be a farness equlbrum, no player can gan utlty by unlaterally devatng from t. Unlateral devaton from a farness eq. mples dong somethng else than your opponent expects. Thus, to check whether the canddate s ndeed a farness eq., we need to compare t wth the alternatves for unlateral devaton, whch are NOT farness equlbra. Let us frst calculate each person s utlty n each of the potental farness eq. To calculate utltes, we must frst calculate kndness and evaluatons of the other s kndness. Is (C,C) a farness equlbrum? Consder persons 1 s perspectve. Assume that 1 thnks 2 wll play Cooperate (b 12 =C) and that 1 thnks 2 beleves 1 wll play Cooperate too (c 121 = C). Then, 2 s kndness to 1, as evaluated by 1, s as follows: k 21= x 1 (b 12, c 121 ) - 1/2[x max 1 (c 121 )+ x mn 1 (c 121 )] = x 1 (C, C ) - 1/2[x max 1 (C)+ x mn 1 (C )] =3-1/2[3+ (-5)]= 3 (-2/2)= 3 +1 = 4 Note that 1 s actual strategy s 1 does not enter the expresson for k 21, only 1 s belefs. When we calculate 1 s actual kndness towards 2, k 12, actual strategy s 1 matters. If 1 keeps to hs (possble) equlbrum behavour,.e. s 1 = C, then (by symmetry) we must have, k 12 = k 21 =4. Insertng ths n the utlty functon, and usng that the game s symmetrc for 1 and 2, we get that the utlty of player =1,2 n (C, C) s U x kjk j Unlateral devaton from a (possble) farness eq. (C, C) for player 1 means to play D even though 2 expects hm to play C. That s, s1=d, b12=c, c121 =C. In ths case, k 12 = x 2 (s 1, b 12 ) - 1/2[x 2 max (b 12 )+ x 2 mn (b 12 )] = x 2 (D, C ) - 1/2[x 2 max (C )+ x 2 mn (C )] = -5-1/2[3+(-5)]= = -4 Insertng ths n the utlty functon, we fnd 1 s utlty by unlateral devaton from (C,C), that s s1=d, b12=c, c121 =C: U1 x1 k12k Thus, devaton from (C,C) decreases utlty for 1. By symmetry the same holds for 2. Hence (C,C) s a farness equlbrum.

7 Is (D,D) a farness equlbrum? If both play D, and beleve that the other plays D, a smlar reasonng gves k 21= x 1 (b 12, c 121 ) - 1/2[x max 1 (c 121 )+ x mn 1 (c 121 )] = x 1 (D, D ) - 1/2[x max 1 (D)+ x mn 1 (D )] =0-1/2[5+0]= (5/2) and agan k 21= k 12 by symmetry. Hence, f both play D and beleve that the other wll play D, the utlty of each player =1,2 s U x k k = 0 +(-5/2)(-5/2)= 25/4 = Ths s a farness eq. f nether 1 nor 2 can gan from unlateral devaton from (D,D). If 1 devates and plays C, whle 2 stll thnks 1 wll play D, k 21 gven by k 12 = x 2 (s 1, b 12 ) - 1/2[x 2 max (b 12 )+ x 2 mn (b 12 )] = x 2 (C, D ) - 1/2[x 2 max (D )+ x 2 mn (D )] = 5-1/2[5+0]= 5 5/2 = 5/2 Insertng ths nto the utlty functon gves U1 = -5 +(-5/2)( 5/2)= = (5/2) as before, whle k 12 s By symmetry of the game the same must hold for 2. Hence (D,D) s a farness eq. Is (C,D) a farness equlbrum? Assume that 1 thnks 2 wll play Defect (b 12 =D) and that 1 thnks 2 beleves 1 wll play Cooperate (c 121 = C). Then 1 s evaluaton of 2 s kndness gves k 21= x 1 (b 12, c 121 ) - 1/2[x max 1 (c 121 )+ x mn 1 (c 121 )] = x 1 (D, C ) - 1/2[x max 1 (C)+ x mn 1 (C )] = -5-1/2[3+ (-5)]= -5 (-2/2)= -5 +1= - 4 What s 1 s actual kndness towards 2? If he plays C (yeldng materal payoffs ( 5,5)), we get k 12 = x 2 (s 1, b 12 ) - 1/2[x 2 max (b 12 )+ x 2 mn (b 12 )] = x 2 (C, D ) - 1/2[x 2 max (D )+ x 2 mn (D )] = 5-1/2[5+0]= 5 5/2 = 5/2 and hs utlty wll be j j 5 U1 x1 k12k21 5 ( 4) If 1 nstead plays D, whle expectatons are unchanged, hs kndness s k 12 = x 2 (s 1, b 12 ) - 1/2[x max 2 (b 12 )+ x mn 2 (b 12 )]

8 = x 2 (D, D ) - 1/2[x 2 max (D )+ x 2 mn (D )] = 0-1/2[5+0]= 0 5/2 = -5/2 and the utlty of player 1 s 5 U1 x1 k12k21 0 ( ) ( 4) Hence (C,D) s NOT a farness equlbrum. Note: I make ths concluson because at least one player (player 1) can beneft from devatng. Thus t s not necessary to calculate utltes for Player 2. Had we found that Player 1 could not beneft from devatng, we would need to check f the same was true for Player 2, because the stuaton (C,D) s NOT symmetrcal. Is (D,C) a farness equlbrum? NO. Short answer, see dscusson of (C,D) above for explanaton: b 12 =C, c 121 =D k = If s 1 =D: k 12 =-4, U 1 =-5 If s 1 =C: k 12 =4, U 1 =13 Summary not a game matrx We may well summarze the payoffs n utltes n all stuatons wth correct belefs, as follows: C D C 19,19-15,-5 D -5, ,6.25 However, ths should NOT be regarded as a fully specfed game matrx. The reason s that some alternatves, namely those choces mplyng that a player thnks he wll surprse the other, are not ncluded n the table. Thus we cannot n general fnd farness equlbra by usng the above table as a standard game and lookng for ts Nash equlbra (although ths would produce a correct answer n ths partcular case); we need, nstead, to consder f players can gan by devatng unlaterally, gven belefs about the others strategy and the others belefs, as demonstrated above.