Tile Values of Information in Some Nonzero Sum Games

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1 lnt. ournal of Game Theory, Vot. 6, ssue 4, page Physca- Verlag, Venna. Tle Values of Informaton n Some Nonzero Sum Games By P. Levne, Pars I ), and ZP, Ponssard, Pars 2 ) Abstract: The paper consders nonzero-sum games n whch the players' utlty functons are not known wth certanty. It tres to clarfy the varous defntons of nformaton value n such games, provdes the reader wth a class of games wth smple solutons, and presents expermental results corroboratng the theoretcal analyss. 1. Introducton Consder nonzero sum games n whch the players' utlty functons are not known wth certanty. Such games are usually modelled as games of ncomplete nformaton [Harsany]. In these games the utlty functons are determned by a chance move, about whch the players are partally nformed. In ths context, the present paper nvestgates the consequences on the players' payoffs, of changng ther state of nformaton on the outcome of the chance move. Tle followng questons wll be debated: 1. Is nformaton always valuable? 2. Is t better to acqure nformaton secretly or n front of others? 3. Is prvate nformaton better than publc nformaton? Informaton value theory s a rather well known subject n classcal decson theory [Raffa]. There, nformaton s always valuable. More recently there has been much research on zero sum ganes of ncomplete nformaton and the basc result for these games shows that nformaton s also always valuable [Ponssard]. Now for non zero sum games, the stuaton seems much more complcated, some studes ndcate that nformaton naay become detrmental [BasarHo] 3). It may also be nterestng to note that much earler Schellng [1960, ] arrved at the same concluson but n a somewhat dfferent context. He nvestgated the problem of communcaton between the players, whereas the subject consdered here s a problem of nformaton about some ncertan event dependng on nature. The contrbuton of ths paper may be seen as follows: 1 ) Prof. P. Levne, Pars, Unverst~ de PARIS VI, Laboratore d'econometre 4, Place usseu, F Pars-Cedex 05. 2) Prof..P. Ponssard, Centre de Recherche en Geston. l~cole Polytechnque, 17, Rue, Descartes, F Pars. 3) In a prvate correspondance Shapley ponted out the fact that n a barganng stuaton the Nash soluton may be more favorable to the players n the presence of an ncertan event than f they become aware of the outcome of ths event,

2 222 P. Levne, and.p. Ponssard () It tres to clarfy the varous defntons of nformaton value for decson problems wth multple decson makers. () It provdes the reader wth a class of games wth smple solutons 4) n whch all possbltes about the value of nformaton may occur. () It presents expermental results whch corroborate the theoretcal analyss. 2. Prelmnary Consderatons on the Value of Informaton To study nformaton problems n compettve stuatons, we shall dstngush dfferent types of nformaton that one player may receve n a game. Case 1: "secret" nformaton In ths case, one taayer acqures the nformaton, but the other players are gnorant of ths fact. Note that the resultng stuaton cannot be analyzed as a game snce the rules are not known to all players. Nevertheless one may assume that the unnformed players wll not modfy ther behavour. Ths assumpton could be partcularly unreasonable f the game was dynamc (n extensve form), snce then the unnformed players would soon realze that they are playng a dfferent game. Case 2: "prvate" nformaton In ths case, one player acqures the nformaton and, though he s the only one nformed, ths fact s known to all players. Then, nformaton may have several effects whch cannot take place wth secret nformaton. Frst, the acquston of nformaton may gve the opportunty to the nformed player to use threats aganst the unnformed players wthout even usng hs nformaton as such. Second, the unnformed players may modfy ther own behavour and t s not clear whether ths would beneft the nformed player. Case 3: "publc" nformaton In ths case, all players acqure the nformaton and ths s known to everybody. At frst sght, one could expect that secret nformaton would be more valuable than prvate nformaton, whch n turn, would be more valuable than publc nformaton s). But ths s not true. The followng class of games shows that one may have all possble confguratons. 4) That s usng only domnance consderatons. Thus our conclusons cannot be attrbuted to the weakness of the soluton concept. 5) Ths s of course the case n zero sum games as a drect consequence of the monotoncty of the value wth respect to the strategy space.

3 The Values of Informaton n Some Nonzero Sum Games The Values of Informaton n a Smple Class of Games 3.1 One Shot Games Consder the followng class of two-person games wth ncomplete nformaton: chance move prob = 12 ~_._....~.~ prob = 12 a a ~ b 3 [(a12, a21) (a22, a22) 3[(a11, a22) (a21 a22) A1 A2 Chance selects one of the two bmatrces A1, A 2, each one wth probablty 12. The players' states of nformaton about the outcome of ths chance move wll be vared. Then, player 1 and player 2 have the choce between two actons (a, 3) and (a, b) respectvely. Note that each bmatrx s symmetrc (the players are nterchangeable) and that one goes from one bmatrx to the other by permutng the two columns and the two rows. Furthermore, the followng restrctons on the parameters are made: () all >a12 and a21 >a22, () all +a12 >a22 +a21, () a12 +a2x >all +a22. Note that ths mples all >a12 >a22. These assumptons greatly smplfy the analyss of ths class of games. Consder the game n whch no player s nformed of the outcome of the chance move. Then, the obvous soluton s (~ a) snce the game s a pure coordnaton problem wth (a, a) domnatng any other outcome. Each player's expected payoff s (all + a12)2. Usng (), n the bmatrx A 1, a domnates3 and a domnates b; whereas n A 2,3 domnates a and b domnates a. Ths solves the game n case of publc nformaton on the outcome of the chance move. Each player's expected payoff s (all + a21 )2. If player 2 s nformed of the outcome of the chance move, then he should play the strategy a n A 1 and b n A 2 (by ()). Now f ths nformaton s acqured secretly, player l's choce remands c~ and player 2's expected payoff s al 1. If t s acqured prvately, player l's best choce s 3, (usng ()), and player 2's expected payoff s a21 9 Then, the value of publc nformaton for player 2 s (a21 -al 2)2. Hs value of secret nformaton s (al 1 -- a12)2 and hs value of prvate nformaton s a21--(all +a12)2.

4 224 P. Levne, and.p. Ponssard Player 2's expected payoffs 6 prvate nformaton publc nformaton secret nformaton.no nformaton I I 3 4 ~ 16 a22 q a12 all I a21 a21 Fg.: Player 2's expected payoffs dependng on the way he acqures nformaton n the followng game 3, a21 1,1 4,1 a21,a2~ Numercal values of parameters for ths fgure: a,, =4;a,2 = 3;a2~ = 1;2=(aa~ +a22)--aa~ <a2~ <~(atl +a~2)--a22 =6 The graph above depcts the respectve postons of these values when the parameter a21 s allowed to vary nsde the constrants mposed by (), (), and (). Consequently, t s possble that the value of prvate nformaton s negatve, or smaller than the value of publc nformaton; t s also possble that secret nformaton s less valuable than ether prvate or publc nformaton. Snce the ranges of a21 for whch these values are postve or negatve do not seem to have ntutve nterpretatons there s lttle hope that a classfcaton of games wth respect to nformatonal propertes can be made.

5 The Values of Informaton n Some Nonzero Sum Games 225 The case n whch the value of prvate nformaton s negatve deserves some more thought. Player 2 should not have acqured ths detrmental nformaton. But f player 1 beleves player 2 has ths nformaton, then the noncooperatve assumpton of the analyss leaves no way for hm to demonstrate that he wll not use t. Remark t s even possble that player 1 may beneft from ths stuaton, and so, has no ncentve to play as f player 2 were not nformed. Here s such an example: chance move prob = 12 ~ ~ prob = 12, a ---~b ~ b a[(2, 2) (2,2) I a[ (2,2) (2,2) 3[(3,1) (0,0) 13 (0,0) (3,1) A1 If nobody s nformed the expected payoffs are (2,2) for players 1 and 2 respectvely. Now f player 2 s prvately nformed, then, snce a domnates b n A 1, and b domnates a n A 2, the payoff becomes (3,1). It may be noted however that, for the class of symmetrc games consdered n ths paper, f the value of prvate nformaton s negatve for the nformed player, that s a21 - (all + a x2)2 < 0, the non-nformed player cannot beneft from the stuaton: Player l's payoff n the game n whch player 2 uses hs prvate nformaton, (a12 + a21 )2, s smaller than (all+ a12)2, hs payoff n the game n whch nobody s nformed, snce a21 <all. Ths suggests that f such a game was terated, then player 2 could play as f he was not nformed and player 1 would not object. Ths s formally studed n the followng secton. Remark 1. Our specal class of games prevents us to observe more complcated features related to soluton concepts. Let us gve an example: The payoffs are gven by: chance move A2 prob = 12 ~ a a[ (0,9) (--1,-4)]! 3 [(-1, 2) (3,-3) A1 prob = 12 a ~!z (-1,-4) (3, 3) As

6 226 P, Levne, and.p. Ponssard Thus, when nobody s nformed, the expected payoffs are: a b (0,3) (-1,-1)] 3[(-1,-1) (3,0) If the second player s nformed the strategy "a n the game A 1 and b n the game A2" s a domnatng strategy for hm. Hence, playe~ I has to play3, and the expected payoffs are (1, 52). But, n the game where nobody s nformed, there are two Nash equlbrum ponts (a, a) and (r b), whch are not nterchangeable. If we assume that, t s (a, a), whch wll be actually played the value of prvate nformaton s negatve for player 2. But, f t s (~, b), ths value s postve. 3.2 Iterated Games We shall assume n ths secton that the games of the class defned n 3. l are not played once, but are terated an nfnte number of tmes. Thus, we shall consder now the supergames [Aumann; LuceRaffa] assocated wth the games of 3.1. The payoffs n these supergames wll be the average of the one shot payoffs. We shall cati value of prvate nformaton for player 2 ~ the supergame, the dfference between : - hs expected payoff n the supergame assocated wth the game n whch he s prvately nformed and - the expected payoff he obtans n the supergame assocated wth the game n whch nobody s nformed. Let us recall that the set of payoffs of the supergame assocated wth a bmatrx game (B = II bff H, C = II c] I[) s precsely the convex hull H of the vectors (b], Cj ) of R 2. For an terated bmatrx game (B, C) t s well known that an outcome belongng to the Pareto boundary of H can be acheved not only under cooperatve behavour but also under noncooperatve beha'aour usng approprate retalatory strateges n case of devaton [Aumann; LuceRaffa]. Then, f one consders only such outcomes for the class of games defned n 3.1 : Theorem: The value of prvate nformaton for player 2 n the supergame s postve. Proof: a) If n the one shot game, the nformaton of player 2 s valuable for player 1 (.e. al 1 < a21 ), one cannot fnd a par of pure strateges of the one shot game n whch player 2 s nformed, such that the correspondng expected payoff for player 1 s greater than (al 2 + a2~)2. Thus there s no Pareto pont of the convex hull H of the expected payoffs of the one shot game whch gves less than (al 2 + al; )2 to player 2, hs payoff n the non-nformed case.

7 The Values of Informaton n Some Nonzero Sum Games 227 b) If for the game played once, nformaton s not valuable for player 1, (.e. al >~a2~ ), one cannot fnd a par of pure strateges of the one shot game such that the correspondng payoff for player 1 s greater than (a~ + a12)2. Thus, there s no Pareto pont of H whch gves less than (all + a12)2 to player 2. Remark 2. It s easy to check that, when nformaton s valuable n the one shot game for player 2, the strateges: "play3 at each stage", and "play a n A ~ and b n A2 at each stage" are Nash equlbrum strateges of the supergame whch are Pareto optmal. Moreover "play a n A 1 and b n A 2 at each stage" s also a MaxMn strategy for player 2. Thus, t s lkely that, when nformaton f valuable for player 2, the players wll play n the supergame as n the one shot game. But f nformaton s not valuable for player 2,'t s not valuable for player 1. In ths case t s lkely that they wll forget the nformaton by playng: for player 1, "a as long as player 2 plays a, otherwse play3" and "a as long as player 1 plays c~, otherwse play a n A :, and b n A 2 " for player 2. These strateges are n equlbrum and are Pareto optmal. Remark 3. Theorem 1 reles manly on tle symmetry property of the class consdered. If symmetry s not satsfed, as n the followng example, the value of prvate nformaton for player 2 n the supergame, may stll be negatve or null, whatever the Pareto pont chosen as soluton of the supergame. The game s: chance move prob = 12 ~ ~ r o b = 12 a --~bb a ~-- b 3[(3,1) (0,0) 13 [(0,0) (3,1) AI A2 Here, every pont of the Pareto boundary of the convex hull of (0, 0), (2, 2), (32, 12) and (3,1), gves a payoff to player 2 less than or equal to 2, whch s hs payoff when he s not nformed. 4. Some Expermental Results A two-person game was expermented among a populaton of 25 students. Only two cases were studed: the case where no player s nformed and the case where player 2 s prvately nformed. Each student receved a descrpton of the rules of the game wth the specfcaton of the player he should act for. He had to play the experment alone. Hs opponent was

8 228 P. Levne, and.p. Ponssard not physcally determned. For each case, he was gven the choce of selectng one pure strategy, or several ones among whch he would be ndfferent (no probabltes ~tatements were asked for). Ths experment may be consdered as a test for the soluton concept used n ths paper. Namely, player 2 has a dfferent domnatng acton dependng on the chance outcome. It should be noted that the value of prvate nformaton for the game chosen s negatve. We expected that the students would all play the theoretcal soluton. The results of the experment are presented as follows: the statstcs of the selected pure strateges are gathered over all the students representng the same player. Then, the frequency of an outcome (, ]) of a game s computed by multplyng the observed frequences of the pure strateges and ]. Ths s compared to the probablty of the outcome (,') as t would result from the theoretcal soluton. The game s: prob = 12 chance..._.~rob = 12 a b a ~ b o~ I (0, 4) (4, 0)] or[(1,1) (1,1) 3 [(4,0)(0,4)1 3 (1, 1) (1, 1) AI A2 theoretcal soluton a) no nformaton: a expermental results b a b 1 I0,4 0, [ 0,02 0,02 b) player 2 prvately nformed: Ot a b a b a b a ~176 1 ~ o011 [ [0,73 0,03 0,03 A1 A2 A1 A2 b 0,23 } 0,73 The followng observatons may be made about the expermental results: Almost all "player 2" students played ther domnatng strategy; however about 13 of "player 1" students dd not change ther behavour n spte of the nformaton revealed to player 2. They dd not seem to understand that ths new nformaton was lkely to modfy player 2's strategy. These partal results ndcate that, even n ths smple compettve stuaton, the students dd not make a sharp dstncton between secret and prvate nformaton and were somewhat confused.

9 The Values of Informaton n Some Nonzero Sum Games 229 Ths remark may explan the answers to a questonnare handed out to the students after the experments. About 34 of them consdered that the value of prvate nformaton n compettve stuatons s always postve. References Aumann, R~ : Acceptable Ponts n General cooperatve n-person Games. In: Contrbutons to the Theory of Games, vol. IV, ed. by A. W. Tucker, and R.D. Luce. Prnceton, N Basar, T., and Y. Ho : Informatonal Propertes of the Nash Solutons of Two Stochastc Nonzero- Sum Games. ournal of Economc Theory, 7 (4), Harsany,. : Games of Incomplete Informaton Played by "Bayesan Players". Part I-H-Ill. Management Scence 14 (3-5-7), Luce, D., and H. Raffa: Games and Decsons. New York Ponssard,.P : On the Concept of the Value of Informaton n Compettve Stuatons. Management Scence 22 (7), Raffa, H. : Decson Analyss. Readng, Mass., Schellng, T.C. : The Strategy of Conflct. Harvard Unversty Press, Receved May, 1976 (revsed verson December, 1976)

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