Formation of Coalition Structures as a Non-Cooperative Game

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1 Formaton of Coalton Structures as a Non-Cooperatve Game Dmtry Levando Natonal Research Unversty Hgher School of Economcs, Moscow, Russa dlevando@hse.ru Abstract. The paper proposes a lst of requrements for a game able to descrbe ndvdually motvated socal nteractons: be non-cooperatve, able to construct multple coaltons n an equlbrum and ncorporate ntra and nter coalton externaltes. For ths purpose the paper presents a famly of non-cooperatve games for coalton structure constructon wth an equlbrum exstence theorem for a game n the famly. Few examples llustrate the approach. One of the results s that effcency s not equvalent to cooperaton as an allocaton n one coalton. Further papers wll demonstrate other applcatons of the approach. Keywords: Non-cooperatve games, Nash equlbrum, cooperatve games 1 Introducton There s a conventonal dchotomy n exstng game theores: the cooperatve game theory CGT versus the non-cooperatve game theory NGT. CGT deals wth coaltons as elementary tems, NGT deals wth strategc ndvdual behavor. Comparson of the game theores s n Table 1. The enlsted propertes are mportant for studyng non-cooperatve fundamentals of ndvdually motvated socal behavor. Table 1. Comparson of game theores CGT NGT type of a mappng a fnte set to a pont a vector to a vector ndvdual acton no always ndvdual motvaton no always explct allocaton of players over coaltons yes no nter-coalton externaltes sometmes vague ntra-coalton externaltes no always, but wth vague coalton defnton Cooperatve game theores are based on the noton that a coalton, a subset of all players, moves as a whole unt only. Ths approach seems somewhat

2 smplfed. Usually economc agents are self-nterested, and do not necessarly make the same move even beng together. From another sde non-cooperatve approach s based on a noton that every player s able to make ndvdual decsons, but there s a low nterest to parttonng players nto coaltons or coalton structures. 1 Studyng formaton of only one coalton from a non-cooperatve game, was ntroduced by Nash, [6], and s known now as Nash Program, Serrano, [8]. Practce of socal analyss and socal desgn requres studyng non-cooperatve and smultaneous formaton of multple coaltons, or coalton structures. Ths makes Nash program be too restrctve. Importance of studyng externaltes between coaltons, a case mpossble wthn the Nash Program, was mentoned by Maskn, [4]. The man complcaton for a non-cooperatve coalton structure formaton stems from the necessty to descrbe complex nter and ntra coalton nteractons whle formng multple coalton structures. Cooperatve game theory equlbrum concepts also the strong Nash equlbrum evade such ssues. Consder an example wth 10 members n a commttee, whch make the grand coalton. Let there are 5, who do not agree and devate from some jont agreement. Do they have the same reason and devate as one group, fve dfferent reasons and devate ndependently n fve groups, two reasons and devate n some groups of two and three, etc? And every devator may have more than one devatng strategy nsde a devatng group. Devatng groups may have nternal conflcts, there could be conflcts between those who stay and those who devate, or between devators from dfferent groups. These questons are dfferent from those addressed by Aumann [1] n the strong Nash equlbrum and by Bernhem et all [2] n the coalton proof equlbrum. The central dea of the presented model s to ncorporate possble cases of devaton nto ndvdual strategy sets of players. Ths s done by parametrzng possble coalton structures by maxmum coalton sze. A constructed game satsfes propertes n Table 1, and has an equlbrum n mxed strateges. As a result t s possble to study devatons of more than one player. The game dffers from those exstng n CGT and NGT. Varyng a maxmum coalton sze we construct a famly of games. Secton 2 contans an example of a game, Secton 3 presents a mechansm to construct such games. The last Secton 4 shows an example wth preferences over coalton structures for an equlbrum refnement. The proof s n Appendx. 2 Example There are 2 players, = 1, 2. They can make choces over a desrable coalton structure, ndexed as a =, f a player chooses to be alone, and ndexed t =, f a player chooses to be together. 2 A player has two strateges NC 1 The terms a coalton structure and a partton of players are consdered as synonmes. 2 An example wth preferences over coalton structures n n Secton 4.

3 and C for every coalton structure. 3 A set of strateges of s S K = 2 = {NC a, C a, NC t, C t }, = 1, 2. Table 2 presents strateges and outcomes of the game. A cell contans a payoff profle and a fnal coalton structure. Every strategy profle s mapped to the par: a payoff profle and a coalton structure. In the equlbrum a player uses mxed strateges C a, C t wth equal probabltes. There are 4 equlbra outcomes wth payoff profles 2; 2, all beng neffcent. The example uses a unanmous rule for coalton formaton, the grand coalton can be formed only f both choose t. Otherwse a player obtans the sngleton coalton. From twelve possble strategy profle only four result n the grand coalton. Table 2 presents a game wth two types of externaltes, nter-coalton and ntra-coalton n. So the example satsfes all desrable propertes from Table 1. Table 2. Payoff for the unanmous coalton structure formaton rule: the same effcent payoffs 0;0 n dfferent parttons. What s a cooperaton then? NC 1,a 0;0 {{1},{ 2} } N 1,a 3;-5 NC 2,a C 2,a NC 2,t C 2,t -5;3 0;0-5;3 {{1},{ 2} } 2; 2, 3;-5 2; 2 0;0 NC 1,t {{1},{ 2} } 3;-5 C 1,t -5;3 2; 2 0;0 {1, 2 } 3; 5 5; 3 2; 2 For the coalton structure wth a maxmum coalton sze K = 1 every player has two equlbrum strateges. A respectve equlbrum s marked wth an upper star, and t s neffcent. For the coalton structure a player has two more equlbrum strateges. Fnal equlbra are marked wth two upper stars,. The equlbra belong to dfferent coalton structures, what make ths game be a stochastc game, Shapley, [9]. Stochastc property appears here not from outsde shock, but from strategc actons of the players. The presented example let players thnk that they can devate ether alone or together. If both can devate, then has and antcpate another has also the followng strateges: a move wthn the same or a dfferent coalton structure wth strateges NC or C for each case. Every such possblty s n ndvdual strategy set S. 3 For smplcty of the motvatng example players do not have preferences over coalton structures, used n example wth ntrovert and extravert players further.

4 3 A game A game s parametrzed by a number K, K = 1,..., N, a sze of maxmum coalton sze n any feasble coalton structure. The parameter K has two meanngs: no more than K agents are requred to make ths largest coalton, and at the same ths coalton needs no more than the same K agents to dssolve. Every player has a set of strateges for every feasble coalton structure. A strategy profle of all players s mapped nto a coalton structure an allocaton of players nto coaltons and a payoff profle for all players. Payoffs are defned separately for every feasble coalton structure. Defnton 1 a smultaneous coalton structure formaton game. A non-cooperatve game for coalton structure formaton wth maxmum coalton sze K s Γ K = N, {K, PK, RK}, S K, U K N, where {K, PK, RK} - coalton structure formaton mechansm, K s maxmum coalton sze, PK s a famly of allowed coalton structures, RK s a famly of coalton structure partton rules, S K, U K N are propertes of players ndvdual strateges and payoffs, such that: N S K RK { PK, U K = {U P : P PK} N } Every ndvdual strategy set S K s a topologcal space, and every famly of utlty functons U K s ntegrable and bounded over N S K.. Defnton 2 an equlbrum n a game Γ K. For a non-cooperatve game Γ K a mxed strateges profle σ K = σ K s an equlbrum f for every subset nk from N, wth a sze 1 k K, and for every player nk a devaton from an equlbrum, σ K σ K, does not generate an ndvdual gan: σ K, σ K EU Γ K σ K, σ K. EU Γ K Defnton of an equlbrum makes a clam only about strategy profles, but not about resultng coalton structures. Theorem 1 exstence. An equlbrum for a game Γ K exsts for any K = 1,..., N. In short, a proof follows from standard propertes of fnte number of bounded ntegral operators wth a fnal applcaton of a fxed pont theorem. If every S K s a topologcal space, then we can always defne a dual space of probablty measure S on t, S s a set of mxed strateges, wth a weak convergence. Then for every N expected utlty operator EU Γ K s bounded, contnuous and compact. Fnally a fxed pont theorem s appled. Complete proof s n Appendx. N

5 An ncrement n K generates a famly of nested games wth nested components: P K P K + 1, S K S K + 1, U K U K + 1, for N and every K = 1,..., N 1. Defnton 3 famly of nested games. A famly of games G = {Γ K: K = 1,..., N} s nested f : Γ K = 1... Γ K... Γ K = N,.e. for any two games Γ K and Γ K + 1 there s P K P K + 1, R K R K + 1, S K S K + 1, U K U K + 1, for every N and every K = 1,..., N 1. Clearly, every game n a famly has an equlbrum n mxed strateges 3.1 Why not to use a threat, Nash 1953 In the paper on cooperatve behavor Nash 1953 offered to used a threat as a basc concept for coalton formaton analyss. Further development of ths concept was transformed nto a blockng coalton Aumann, The reason not to take them for non-cooperatve coalton structure formaton s the followng. Consder a strategy profle from only part of players. Let ths profle be a threat to someone, beyond ths subset. The threatenng players may produce externaltes for each other and negatve externaltes not excludng!. How credble could be such threat? From an other sde, there may be some other player beyond the subset of players who may obtan a bonanza from ths threat. But ths benefcary may not jon the group due to some ntra-group negatve externaltes for members or from members of ths group. Resultng nternal complexty and recursve nature of the example was the reason not to use a threat and blockng coalton concepts to establsh non-cooperatve formaton of coalton structures. 4 Equlbrum refnement n the example Non-cooperatve game theory extensvely studed refnement of the Nash equlbrum. We can demonstrate equlbrum refnement for the game above wth an ntroducton of ndvdual preferences over coalton structures. Let player = 1 be an extrovert and always prefers to be together, n, but player = 2 s an ntrovert and always prefers to be alone. Player = 1 has an addtonal gan from beng together n comparson to Table 2 ϵ, 0 < ϵ, from beng together wth another. From another sde player = 2 has an addtonal gan from beng alone, δ, 0 < δ. Payoffs of the Table 3 are adjusted accordng to these assumptons and present the payoff matrx for the new game. Usng the rule of unanmous agreement for coalton formaton the grand coalton,, can never be formed, what leaves only one equlbrum payoffs 2; 2 for the game n.

6 Players dffer n number of equlbrum strateges. Player = 2 has one equlbrum strategy C 2,a. Player = 1 has two pure equlbrum strateges, C 1,a and C 1,t and mxes them wth equal weghts. Equlbrum for K = 1 s labeled wth one star,, for K = 2 wth two starts, Table 3. Payoff for a game when player 1 s extrovert and player 2 s ntrovert. NC 2,a C 2,a NC 2,t C 2,t 5; 3 + δ 0; 0 + δ 5; 3 + δ 2; 2 + δ, 3; 5 + δ 2; 2 + δ NC 1,a 0; 0 + δ C 1,a 3; 5 + δ 0; 0 + δ NC 1,t 3; 5 + δ C 1,t 5; 3 + δ 2; 2 + δ 0 + ϵ; ϵ; ϵ; 2 + ϵ; 2 If both players are extroverters wth a premum ϵ > 0 for beng together n, then a change n a game from K = 1 to K = 2 changes the equlbrum strategy profle, and a resultng equlbrum coalton structure as well. Table 4 contans payoffs for ths game. The addtonal payoff makes players form only the grand coalton wth a unque equlbrum, but a resultng payoff s stll neffcent. Table 4. Payoff for two extrovert players who prefer to be together. Unqueness of an equlbrum s fxed. L 2,a H 2,a L 2,t H 2,t -5;3 0;0-5;3 2; 2 3;-5 2; 2 L 1,a 0;0 H 1,a 3;-5 L 1,t 0;0 H 1,t 3;-5-5;3-2; ϵ; 0 + ϵ 3 + ϵ; 5 + ϵ 5 + ϵ; 3 + ϵ 2 + ϵ; 2 + ϵ 5 Concluson The paper suggest a new famly of smultaneous non-cooperatve games whch satsfy the followng crtera: be non-cooperatve, be able to form multple-

7 coalton structures and to contan two types of externaltes, ntra and nter - coalton. Further applcaton of the proposed mechansm to follow n the next papers. Earler verson of the paper wth more applcatons was publshed at SSRN. Acknowledgments. Specal thanks for the support to Fuad Aleskerov, Lev Gelman, Dmtros Tsomocos, Marc Kelbert, Nadezhda Lkhacheva, Olga Pushkarev, Shlomo Weber and some others. Earler verson of the paper was publshed at SSRN and at arxv. References 1. Aumann, R.Y., Acceptable ponts n games of perfect nformaton. Pacfc Journal of Mathematcs, 102, Bernhem, B.D., Peleg, B. and Whnston, M.D., Coalton-proof nash equlbra. concepts. Journal of Economc Theory, 421, Lyusternk, L.A. and Sobolev, V.I., A short course n functonal analyss. Vysshaya Shkola, Moscow Maskn, E., Commentary: Nash equlbrum and mechansm desgn. Games and Economc Behavor, 711, Nash, J., Non-cooperatve games. Annals of mathematcs, Nash, J., Two-person cooperatve games. Econometrca: Journal of the Econometrc Socety, Schllng, R.L., Measures, ntegrals and martngales. Cambrdge Unversty Press, Serrano, R., Ffty years of the Nash program, , Shapley, L.S., Stochastc games. Proceedngs of the natonal academy of scences, 3910, Appendx. Proofs Theorem 1. Let K be a maxmum sze of a coalton. For fxed K a game Γ K has the Nash equlbrum n mxed strateges. The parameter K s omtted everywhere below for smplcty, besdes expected utlty. Step 1 Prelmnary step. Let S be a topologcal space for every N. Let BS be σ-algebra on S, and the par S, BS be a measurable set. Let a non-negatve measure on S be a mappng : BS [0, 1], and S, BS, S be a measure space of mxed strateges wth S = 1. By theorem 5 from [7] the space can be unquely assgned. The space S, BS, has weak convergence, what we do not prove here. Lemma 1. The space S, BS, S s complete.

8 { Proof. Let σ, be a mxed strategy, σ, and } be a sequence of mxed strateges of. All the sequences n S, BS, are bounded. By Bolzano-Werstrass theorem every bounded { sequence } has a least one lmt pont. Let σ 0 be a lmt pont of a sequence, and σ 0 s ds = lm S k s ds. S Exstence of a lmt pont n the weak convergence sense follows from the weak compactness property of the set, by Prohorov s theorem, [7]. From ths mmedately follows completeness of the set of mxed strateges. A set of probablty measures := j j of all other players besdes s bounded. Let σ be a mxed strategy of all other players, σ. Step 2 Expected utlty of one player. Let U : N S R + be a payoff functon of, contnuous on S S and U S S s, s 2 ds ds <. We wll use the word operator to emphasze, that player controls only own strateges, varables ndexed wth, but does not control varables ndexed wth controlled by other players. Expected utlty operator of n a game Γ K s a Lebegue ntegral EU K σ, σ := EU Γ K σ, σ = S S U s, s σ s σ s ds ds. Probablty measures for all players and each partton-contngent payoffs U s, s were defned above, so expected utlty EU K σ, σ s well defned. Theorem { 2. Expected } utlty operator EU K σ, σ maps a weakly convergng { } sequence nto a bounded convergng sequence EU K, σ. Proof. Let lm k S s ds = S σ 0 weakly convergng sequence from. Then lm k EU K = { s ds, where } s a bounded, σ = lm U s, s s σ s ds ds k S S U s, s S S s σ s ds ds = EU K σ 0, σ, Expected utlty EU K over S. σ, σ nherts compactness and weak convergence

9 { Lemma 2. The sequence EU K. }, σ s contnuous on, σ Proof. lm k EU K, σ EU K = lm U s, s k S S U s, s ds ds lm S S k σ 0, σ s σ 0 s σ s ds ds S S s σ 0 s σ s ds ds = 0 Let F σ = arg max σ EU K σ, σ, F :. Operator F s a best response operator, t s bounded, compact and contnuous. Best response correspondence F determnes the best mxed strateges of to maxmze expected utlty gven mxed strategy profle from other players. Expected utlty operator EU K σ, σ maps a convex, closed, compact set, F, to a convex, closed, compact set. Theorem 3. Let be a set of probablty measures of N, and X = EU K, be a bounded and compact set of s expected utlty values. Then the set F X = arg max σ S EU K σ, σ s also compact. { } Proof. The set X s a compact set. Let be a sequence from. k =1 For every pont we take one of the pre-mages X k, F X k =. The { } set X s compact, then from the sequence X k X we can extract { } a convergng sub-sequence X kr, whch converges to X0 X. kr=1 By contnuty of F X on X at X 0 there s σ kr = F X kr F X 0 = σ 0. Step 3 Fxed pont argument for the game To apply fxed pont theorem we construct a system of N equatons and demonstrate exstence of a fxed pont for the whole system. Let a vector operator EU K σ = EU K σ, σ be a lst of expected utlty operators of all players. By the Tykhonov theorem the set of expected payoff N values EU K, s.t. =, s compact, = j. Thus the set EU K s also bounded and contnuous over the set of probablty measures, as every component of EU K, has ths property. Let k be a number of a probablty measure mxed strategy of n a sequence of {{ probablty measures } } N n the sense of the weak convergence, 1

10 weak s.t. σ 0. We want to construct a fnte collecton of convergng expected utlty operators. For every exsts an argmax operator: F σ :,.e. the best response operator, that allows to choose the best mxed strategy from n response to σ from, such that F σ = arg max σ EU K σ, σ, where s a bounded, closed, convex set wth weak convergence. Let { F k { } = arg max be a sequence of vectors of operators. Let F k 1,..., F k N arg = EU K max EU1 K 1, σ 1,..., arg max 1 1 N, σ N } EU K N N, σ N 2 be a sequence of best response operators. Every vector operator F k = s a fnte vector, and by Tykhonov F k N theorem t s also a compact and closed set. Every mappng F k transforms a convex, closed, compact set = N nto tself. Lemma 3 Schauder. Any operator F compact on N has a unform lmt on = N as a sequence of contnuous fnte dmensonal operators F k 1,..., F k N, and maps S, S S, S. The proof follows Sobolev and Ljusternk, Proof. Operator F s a compact operator, hence F s a compact set. Let there s a sequence of postve numbers {ϵ 1, ϵ 2, ϵ k,...}, whch converges to zero. Let y F be an element of. For every small postve ϵ k we construct L k = {y k 1,..., yk N }, yk = y k for every there s y k strategy profle y y k = max N N { 1,..., yk N F, where } N. All L k are contnuous on a mxed n a sense of the weak convergence, and denote S S y k s y s y s ds ds. Defne on F an operator J k such that for any y F there s

11 where θ k y = J k y = =1 θk { ϵ k y y k yy k =1 θk y, y y k < ϵ k 0, y y k. > ϵ k Operator J k y s well defned for every y F as θ k at least for one j. Operator J k y s contnuous on F. Ths follows from that all θ k every y F there s 1 θ k y θk, 0 and θ k > 0 y are contnuous over y, =1 θk y s contnuous over y. For y > 0, from where follows contnuty of y. Then y J k y = y =1 θk =1 θk =1 θk =1 θk y yy k = =1 θk yy y k =1 θk y y y y k =1 ϵ θk y k =1 θk y = ϵ k. =1 θk y If there s y y k ϵ k then a correspondng coeffcent θ k y = 0. Let { F } k σ = J k F σ, then for any σ there s a sequence of operators F k such that for any σ. F σ F k σ = F σ F k F σ ϵ, For every y k there s y k F, so the values of the constructed operators belong to a convex closure of F. Lemma 4. Let a sequence of operators { F k} be contnuous on and t unformly converges to an operator F on. Let L k = F k, k = 1, 2,.... Then the set L = Lk s compact. The proof follows Sobolev and Ljusternk, Proof. Accordng to the Lemma above an operator F 0 s compact, as there s a sequence of operators, { F k}, whch converges to F 0. Then for any ϵ k > 0 and when k k, for every σ L k there s y 0 L 0 such that y y 0 < ϵ k. Ths s possble as y s an element from L k and σ s one of pre-mages of y under the mappng F k. So we can take y 0 = F 0 σ. Now we construct the set k L k. Ths set s compact. We need to show that t s an ϵ-net for L. Let y K. If y k L k, then the result of the Lemma s trval. If y L k for k > k, then there s y 0 L 0, such that y y 0 < ϵ k. Thus k= k+1 Lk s compact for ϵ k from set L, and thus L s compact.

12 Theorem 4 Schauder fxed pont theorem. If a compact operator F maps a bounded closed convex set nto tself, then the mappng has a fxed pont, σ, F σ = σ. The proof follows Sobolev and Ljusternk, [3]. Proof. Take a sequence of postve {ϵ k }, convergng to zero, and as above construct a sequence of contnuous fnte-dmenson operators {F k }, whch unformly converges on to F. The nfnte set of all probablty dstrbutons s convex, {F k } = { } for every σ. Let Xk be a fnte dmenson subspace wth the set F k, F k X k. Take the operator F k on the subset k = F k from X k. The set k s also a closed convex set. As F k and F k k, then F k k, and therefore F k k. Thus the operator F k n k, k, maps a closed convex set k. By the Bauer fxed-pont theorem there s a fxed pont of ths mappng,.e. F k =. From another sde k, thus s a fxed pont also for { the } mappng F k. As the pont F k, and the sequence σ k belongs to the set = F k. Thus the set s compact. Then from the sequence { } we can extract a convergng subsequence { σ kr} k r=1 and a lmt of ths subsequence s n, as s closed. We can defne F kr σ kr, σ 0 = max N S S F kr σ kr σ 0 F kr σ kr = σ kr. We need to demonstrate that σ 0 s a fxed pont. Ths follows from F σ 0, σ 0 F σ 0, F σ k r + F σ k r, F k r σ k r + F k σ k r, σ 0 = = F σ 0, F σ kr + F σ kr, F k σ kr + σ kr, σ 0. σ 0 ds ds, For any fxed ϵ k > 0 we choose k be bg enough that for any k > k there s σ k, σ 0 < ϵ k/3 and F σ 0, F σ k < ϵ k/3. Then we choose k be bg { enough } that for k k there s F σ, F σ k < ϵ k/3 s unform on for all σ k. Then for some k max{k, k } there s k= k F σ 0, σ 0 < ϵ k/3. As ϵ k > 0, then ths s possble only f F ϵ 0 = ϵ 0. Then ϵ 0 s a fxed pont. The proof does not depend on K, thus every game n the famly G has an equlbrum n mxed strateges.