Sequence of opponents and reduced strategies Jean-Pierre Beaud

999²⁰⁰⁰9

Sequence of opponents and reduced strategies

Jean-Pierre Beaud1and Sylvain Sorin1,2

Received: March 1999/Revised: May 2000

Abstract.We consider the framework of repeated two-person zero-sum games with lack of information on one side. We compare the equilibrium payo¨s of the informed player in two cases: where he is facing either a) a single long- lived uninformed player, or b) a sequence of short-lived uninformed players.

We show: 1) that situation b) is always (weakly) better than a), 2) that it can be strictly better in some cases, 3) that the two cases are equivalent if the long uninformed player has an optimal strategy independent of his own moves.

Key words:incomplete information, sequence of opponents

1. Presentation

We consider a two person zero-sum repeated game with incomplete informa- tion on one side ([1]). There is a ®nite state spaceK. For eachkAK, aIJ real payo¨ matrix A^k is given, as well as two IJ signalling matricesH^1;k andH^2;k with values in ®nite signal setsSandT.

The game G is played as follows: k is selected according to a publicly known distribution p. Player 1 (the maximizer) is informed on the outcome but not Player 2. Inductively at stage m, Player 1 chooses an action im in I and Player 2 chooses an action j_minJ. The stage payo¨ isxmˆA_i^k_mjm but is not announced. Rather the signal s_mˆH_i^1;k_mjm (resp. t_mˆH_i^2;_m^k_jm) is told to Player 1 (resp. Player 2). We assume that for each player his signal contains

1 Laboratoire d'EconomeÂtrie, Ecole Polytechnique, 1 rue Descartes, 75005 Paris, France he-mail: beaud@poly.polytechnique.fri

2 MODALX and THEMA, UFR SEGMI, Universite Paris X, 200 Avenue de la ReÂpublique, 92001 Nanterre, Francehe-mail: sorin@poly.polytechnique.fri

his move. A behavioral strategy for Player 1 is described by a sequence sˆ fs_mgof mappings fromKS^{m 1}to the set of mixed actionsD…I†. Similarly, but taking into account his lack of information, a behavioral strategy of Player 2 is a sequencetˆ ft_mg of mappings from past signalsT^{m 1} toD…J†.

Denote bySandTthe corresponding sets. Any triple…p;s;t†inD…K† S T induces a probability distribution on the set of plays K …ST†^y and Ep;s;tdenotes the induced expectation.

The game G is played the same way as the game G but with a di¨erent Player 2 at every stage. However each of these players knows the history of signals sent to his predecessors. We call 2:mthe Player 2 active at stage m, hence his strategy's set is also described by a mappingy_mfromT^{m 1}toD…J†.

For each class of games we consider the ®nitely repeated and discounted version:

Gn…p† is then stage two person zero sum game with payo¨ for Player 1 given byg_n¹…p†…s;t† ˆE_p;s;t 1

n X_n

mˆ1x_m

.G_n…p†has a valuev_n…p†.

G_l…p†is thel-discounted two person zero sum game with payo¨ for Player 1 given byg¹_l…p†…s;t† ˆE_p;s;t…lP_y

mˆ1…1 l†^{m 1}x_m†.G_l…p†has a valuev_l…p†.

Gn…p† is a …n‡1† player game with payo¨ g_n¹…p†…s;y† ˆg_n¹…p†

…s;y₁;. . .;y_n† for Player 1 and g^2;m…p†…s;y† ˆE_p;s;y… x_m† for Player 2:m,

mˆ1;. . .;n. (Note that this last payo¨ depends only upon …sl;yl† for

lUm).E_n…p†denotes the set of equilibrium payo¨s of Player 1 inG_n…p†.

Gl…p† is a game with an in®nite family of Player 2. The payo¨ is g¹_l…p†…s;y†for Player 1 and againg^2;m…p†…s;y† ˆE_p;s;y… x_m†for Player 2:m, mV1. The existence of equilibria inG_l…p† follows from standard approxi- mation by truncated games with ®nitely many Player 2. LetE_l…p†denote the corresponding set of equilibrium payo¨s for Player 1.

The purpose of this note is to comparevn…p†andEn…p†(respectivelyvl…p†

andE_l…p†), namely to examine whether it is pro®table or not for the informed player to face a single long lived opponent or a sequence of short lived oppo- nents.

2. A ®rst result

Player 1 is always better o¨ facing a sequence of short lived opponents since we have:

Proposition 1.Ep;EnV1;EyAE_n…p†: yVvn…p†

SimilarlyEp;ElA…0;1†;EzAEl…p†:

zVvl…p†

Proof:The proof is the same in both cases.

Player 1 faces the same set of strategies Tin Gand G. Thus the use by Player 1 inG_n…p†orG_l…p†of an optimal strategy inG_n…p†orG_l…p†implies

the result. 9

3. Reduced strategies

De®nition. tis a reduced strategy of player 2 if it depends upon his signals only through the moves of Player 1.

In other wordst_m is measurable with respect to the algebra generated by …i1;. . .;im 1†. Explicitely: say that …i1;. . .;im 1† is compatible with …t1;. . .; tm 1† if there exists k with tlˆH_i^2;k_l;jl for lˆ1;. . .;m 1, (recall that tl

determines j_l). Now if some …i1;. . .;im 1† is compatible both with …t1;. . .; t_{m 1}†and…t₁⁰;. . .;t_m⁰ ₁†thent_m…t₁;. . .;t_{m 1}† ˆt_m…t₁⁰;. . .;t_m⁰ ₁†.

Theorem 2.If Player 2 has a reduced optimal strategy in G_n…p†(resp. G_l…p†), thenEn…p†is reduced tofvn…p†g(resp.El…p†is reduced tofvl…p†g.

Proof: Let …s;y† be an equilibrium in Gn…p†. Let t^? be a reduced optimal strategy of Player 2 inGn…p†. The equilibrium condition for Player 2:mgives:

g^2;m…s;y†Vg^2;m…s;y _m;t^?_m†

The idea is to de®ne inductively a ®ctitious strategy of Player 1,s, as a func-~ tion ofsandysuch that, for allmˆ1;. . .;n:

g^2;m…s;y _m;t^?_m† ˆg^2;m…~s;t^?†:

~

sis a sequence of mapss~_mfromKI^{m 1}toD…I†(hence a strategy of Player 1 independent of the moves of Player 2) de®ned inductively as follows:

~ s₁^k ˆs₁^k

~

s₂^k…i1† ˆ X

s1AS;j1AJ

y1…j₁†1_fH^1;k_i1;j1†ˆs1gs₂^k…s1†

In words, Player 1 is playing at stage 2 the expectation of his initial strategy with respect to the distribution on signals induced byy1. Similarly, at stagem,

~

s_m is de®ned by taking the conditional expectation on the signals of Player 1, given his moves:

~

s_m^k…i₁;. . .;i_{m 1}† ˆE…s_m^k…s₁;. . .;s_{m 1}† ji₁;. . .;i_{m 1}†

ˆ X

s1;...;sm1

P^k…s1;. . .;sm 1ji1;. . .;im 1†s_m^k…s1;. . .;sm 1†

where P^k…s₁;. . .;s_{m 1}ji₁;. . .;i_{m 1}† is the probability (under s^k and y) of s1;. . .;sm 1 giveni1;. . .;im 1(recall thatsdeterminesi).

We now have:

g^2;m…p†…s;y m;t^?_m† ˆX

kAK

p^k X

s1;t1;...;sm 1;tm1

P^k…s1;t1;. . .;sm 1;tm 1†

s_m^k…s₁;. . .;s_{m 1}†… A^k†t^?_m…t₁;. . .;t_{m 1}†

Taking conditional expectation with respect to …i1;. . .;im 1† we have, since t^? is reduced, hence measurable w.r.t. …i₁;. . .;i_{m 1}† (and E^k stands for the expectation w.r.t.s^kandy):

g^2;m…p†…s;y m;t^?_m† ˆ X

kAK

p^kE^k…s_m^k… A^k†t^?_m†

ˆX

kAK

p^kE^k…E^k…s_m^k… A^k†t^?_mji₁;. . .;i_{m 1}††

ˆX

kAK

p^kE^k…E^k…s_m^k ji₁;. . .;i_{m 1}†… A^k†t^?_m†

ˆX

kAK

p^kE^k…s~_m^k… A^k†t^?_m†

Since the distribution of …i₁;. . .;i_{m 1}† is the same under …s;y _m;t^?_m† and …~s;t^?†, we obtain:

g^2;m…p†…s;y m;t^?_m† ˆg^2;m…p†…~s;t^?† It follows that:

ng¹_n…p†…s;y† ˆ Xⁿ

mˆ1

g^2;m…p†…s;y†U Xⁿ

mˆ1

g^2;m…p†…s;y _m;t^?_m†

ˆ Xⁿ

mˆ1

g^2;m…p†…~s;t^?† ˆng¹_n…p†…~s;t^?†Unv_n…p†

Hence

vn…p†Vg_n¹…p†…s;y†

which was to be shown.

The proof is analogous in the discounted case. 9 De®nition.Anon strategic signallingstructure for Player 2 satis®es:

H^2;k…i;j† ˆH^2;k…i⁰;j† ,H^2;k…i;j⁰† ˆH^2;k…i⁰;j⁰† Ej;j⁰

In words, the information that Player 2 obtains on Player 1's move is inde- pendent of his own move. A typical example is standard signalling where H^2;k…i;j† ˆ …i;j†.

Theorem 3.Assume non strategic signalling for Player 2. ThenE_n…p†is reduced tofvn…p†g(resp.El…p†is reduced tofvl…p†g.

Proof:The proof follows from Theorem 2 and the fact that under non strate- gic signalling Player 2 has a reduced optimal strategy.

This property can be obtained using the recursive formula for the (primal) game with incomplete information on one side ([1], [4]): since Player 1 knows the posterior distribution of Player 2, he does not have to know the moves.

The recursive formula for the dual game now implies the result (see [2], [5]).

An alternative direct approach is as follows (see [3], [6]). Assume that Player 1 playss, independent of the moves of Player 2. Then anytwill induce at each stage the same total distirbution on moves than its expectationt⁰with respect to Player 2's previous moves. In particular the payo¨s given…s;t†and …s;t⁰†will coõÈncide. On the other hand, if Player 2 uses a reduced strategyt, the payo¨ of Player 1 is the same for a strategy s or for s⁰, its expectation (givent) with respect to Player 2's moves. Denoting byS⁰andT⁰the corre- sponding sets of reduced strategies one thus has:

minTmax_S⁰ ˆmin_T⁰max_S⁰ min_T⁰maxSˆmin_T⁰max_S⁰ hence:

min_T⁰max_SUmin_Tmax_S…ˆmax_Smin_T†

So that Player 2 has an optimal reduced strategy. 9 Remark:The same result holds if Player 2 receives a random signal indepen- dent of his moves. However in this case Player 1 may be unable to compute the posterior distribution of Player 2 and there is no recursive formula for the value onD…K†.

4. An example

The following example, due to Ponssard and Sorin, was initially constructed to show the impact of the signalling structure on the speed of convergence of the valuesv_n…p†, (see [4] ex. 9, p. 303). The data are:

Iˆ fT;Bg; Jˆ fL;M;Rg; Kˆ fk₁;k₂g

A^k1 ˆ 0 2 2

0 2 3

A^k2ˆ 2 0 3

2 0 2

H^1;k…i;j† ˆ …i;j† (standard signalling). The signalling matrix for Player 2 is deterministic, independent of the state and given by:

H²ˆ a b c a b f

Then one has (with pˆp¹):

v_y…p† ˆ ⁴⁷ on ‰²₇;⁵₇Š, 2 minfp;1 pg otherwise.

v₁…p† ˆ2 minfp;1 pg:

Proposition.For pˆ¹₂and n large enough, there exists an equilibrium…s;y†of G_n…p†satisfying:

g¹_n…p†…s;y†>v_n…p†

A similar property holds forG_l…p†.

Proof: Simply de®nes¹ as always T,s² as always B andymˆ …¹₂;¹₂;0†. The one shot gameG₁…¹₂†is

0 BB B@

L M R

TT 1 1 ¹₂

TB 1 1 2

BT 1 1 3

BB 1 1 ¹₂ 1 CC CA

sis clearly a best reply toy.

At stage m, given s and y, the posterior probability on K is still …¹₂;¹₂†.

Player 2:mis thus facing the gameG1…¹₂†whereRis a dominated move. Hence ymis a best reply andg¹_n…p†…s;y† ˆ¹₂.

Sincevn…¹₂†converges tovy…₂¹† ˆ⁴₇the result follows.

The proof forGlis similar. 9

Comments.A single uninformed Player 2 has a long term incentive to buy the information in playing the move Right. Obviously such a strategy is domi- nated for a short lived Player.

Acknowledgments.We thank the referees for precise remarks.

References

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