Information, Control and Games Shi-Chung Chang EE-II 245, Tel: 2363-5251 ext. 245 scchang@cc.ee.ntu.edu.tw, http://recipe.ee.ntu.edu.tw/scc.htm Office Hours: Mon/Wed 1:00-2:00 pm or by appointment Yi-Nung Yang (03 ) 2655201 ext. 5205, yinung@cycu.edu.tw

Introduction to Cooperative Games Coalitional games

Why do we cooperate? To reach a “win-win” outcome A Pareto improvement Increasing one’s benefit without reducing other’s benefit Different aspects of the cooperation cooperation in the prisoner’s dilemma game:違反社會正義 collusive duopoly: 違反公平交易 (消費者受害) capacity sharing

Coalitional game Terminology A coalitional game consists of Payoffs: n players: indexed by number {1, 2, 3, ..., n} Grand coalition N= {1, 2, 3, ..., n} Any coalition S  N : a subset of the grand coalition A coalitional game consists of a set of players a set of action for each coalition preferences for each player over the set of all actions of all coalitions of which she is a member. Payoffs: vi is the payoffs received by member i who joins N (S) v(N) = iN vi v(S) = iS vi

Two-player unanimity game Scenario 2 players action : Yes/No forming a grand coalition if both say “Yes” => N={1,2} otherwise Si = {i}, i=1,2 payoffs utility v1 = v2 = 1 if 2 players are in the grand coalition vi (Si)= 0 for Si = {i}, i=1,2

Three-player majority game 多數決合作賽局 Scenario 3 players action : forming 3-player coalition: grand coalition N={1,2,3} 2-player coalition: majority coalition S={{1,2}, {1,3}, {2,3}} 1-player coalition S ={{1}, {2}, {3}} payoffs v({i}) = 0 for i = 1,2,3 v(S) = 1 for every other coalition

Landowner and Workers game Scenario m+1 players N={1, 2, ..., m} the landowner is indexed by 1 2, 3, ...m are workers action : A coalition consisting solely of workers (produces NONE) A coalition consisting the owner and n workers payoffs

Redistribution of payoffs Transferable payoff A coalitional game has transferable payoff if there is a collection of payoff functions, every action of S generates a distribution of payoffs among the members of S that has the same sum Two-player majority game non-transferable payoff v1 = v2 = 1 if i is in the grand coalition transferable payoff v(N) = v1 + v2 =2

Payoff Space In a 2-player unanimity game without and with (linear) transferable payoffs

Non-linear transferable payoff Payoff possibility set

3-player payoff space Non-transferable payoffs

3-player payoff space Transferable payoffs v(N) v({2,3}) v({1,2})

3-player payoff space (non-linear) Transferable payoffs v(N) v({2,3})

Payoff space in a simplex form v({1})+v({3})v({1,3}) v({1}) +v({2})v({1,2}) v({2})+v({3}) v({2,3}) v(N) =v1+v3

A game in Characteristic Form The characteristic form is meant to be a summary of the payoffs available to each group of players in a context where binding commitments among the players of the group are feasible Two-player unanimity game V(N) = {v1(aN)=0.5, v2(aN)=0.5: aN=forming a N} V({i}) = 0 for i = 1, 2 if payoffs are transferable 令 xi=vi(aN) V(N) = {x1+x2=1: aN=forming a N}

Cohesive coalitional game A coalitional game is cohesive (內聚性)(superadditive) if the grand coalition N has an action at least as desirable for every player i as the action aSj of the member Sj of the partition to which player i belongs 合比分好 v(N)  kK v(Sk) for every partition {S1, S2,..., SK } of N Examples Two-player unanimity game v(N)  V({1})+v({2})=0

The Core The Core of a coalitional game is the set of actions aN of the grand coalition N such that no coalition has an action that all its members prefer to aN Nash eq. An outcome is stable if no deviation is profitable the core an outcome is stable if no coalition can deviate and obtain an outcome better for all its members The core always exists but could be an empty set

Two-player unanimity game and the core Scenario 若兩人皆同意則 share 1 單位的利益, 否則兩者 payoffs 皆為 0 沒有分配 (division) 的規則 (限制) v(N) = iN xi =1 v({i}) = 0 for i = 1, 2 The core consists of all possible division: {(x1, x2): x1+ x2 =1 and xi  0 for i =1, 2} Only two choice for each i: joint or not joint

Two-player unanimity game If v({i}) = k > 0: Endowment effects

Empty core of the two-player unanimity game Rules of division matter additional constraints x1  p and x2  q Additional constraints in graphical representation

3-player cooperative game and the core Three-player majority game Payoffs v({i}) = 0 for i = 1,2,3 v(S) = 1 for every other coalition grand coalition: v(N) two as the majority: v(M) The core is empty or nonempty? Under equally-shared rule Any one in grand coalition shares 1/3 Any one in coalition M share 1/2 Any two member can exclude the 3rd to improve their share 黑吃黑 =>不穩定集團

n-person majority game The Scenario Let |S| denotes the number of persons in coalition S A group of n player, where n  3 is odd grand coalition: |N| = n A coalition consisting of a majority of the players can divide the unit among its members. The model v(S) = 1 if |S|  n /2 = 0 otherwise The core is Empty?

A modified majority game Three-player unanimity game Payoffs v({i}) = 0 for i = 1,2,3 two as the majority: v(M) =0 grand coalition: v(N)=1 The core is not empty Proposition: there exists a nonempty core with  where v(M) = ,  [0,1] v({i}) = 0 for i = 1,2,3 two as the majority: v(M) =  grand coalition: v(N)=1  = ?

Simplex form for of 3-person cooperative game 以 player 1, 3 為角度來思考 (play 2 gets 0 in N) M={1,3} (v1, v3)| iM = (/2,  /2) N={1,2,3} (v1, v2, v3)| iN = (1/2, 0, 1/2)

3-player cooperative game in the simplex form Majority rule with equal shares v(N)=1, vi = 1/3 for i = 1, 2, 3 v(M)= (此例  = 1/2) M={1,3} (v1, v3)| iM = (/2,  /2) =(1/4, 1/4) Equal shares: N={1,2,3} (v1, v2, v3)| iN = (1/3, 1/3, 1/3) N={1,2,3} (v1, v2, v3)| iN = (1/2, 0, 1/2)

3-player cooperative game in the simplex form Majority rule with equal shares 同樣的想法類推...

3-player cooperative game and the graphics The largest  with non-empty core occurs in the interception of the tree blue lines

3-player cooperative game and the graphics The largest  with non-empty core occurs in the interception of the tree blue lines

Empty and Non-Empty Core in a simplex form

Vote trading game Scenario There are three bill: A, B, and C There are three parties 1, 2, and 3 The numbers indicated in the figure are the party’s payoff For example, if bill A and B pass and C fails, the party 1 gets (2-1)=0 party 2 gets (1+2)=3 party 3 get (-1+1)=0 Is the core empty? (三黨大團結, 3個法案皆通過)

The core of voting trading game

The core of the Landowner and workers game Payoffs Case 1: 3-player (one owner and 2 workers grand coalition: v(N) = f(3) (x1, x2, x3) be an allocation of v(N) (note that 1 is the owner) The core (x1, x2, x3) is in the core if and only if x1 + x2 + x3 = f(3) x1+ x2  f(2) x1+ x3  f(2) x1  f(1), x2  0, x3  0

Analysis of the core The core (x1, x2, x3) is in the core if and only if grand coalition payoffs x1 + x2 + x3 = f(3) (1) deviation from the grand coalition is not profitable x1+ x2  f(2) (2) x1+ x3  f(2) (3) the share of payoffs in grand coalition > stand alone x1  f(1), (4) x2  0, and x3  0

Another Aspect Marginal contribution x1 + x2 + x3 = f(3) ==> x1 = f(3) - x2 - x3 substitute into (2), (3), and (4) (after re-writing) 0  x3  f(3)-f(2) 0  x2  f(3)-f(2) share of the 3rd person  marginal contribution of the 3rd person x2 + x3  f(3)-f(1) share of the 2nd and 3rd person  marginal contribution of the 2nd and 3rd person

Case 2: A group of n  3 in the landowner and worker game Output of k people = f(k), and f(0) =0, Total number of the game: n grand coalition N={1, 2, ...n)} (1 is the owner) An allocation of v(N): x1 + x2 +...+xn = f(n) Under what conditions on (x1 , x2 ,...,xn) is in the core? One (j) deviates from the N and his share is xj output: f(n  1) x1 + x2 +...+xn xj  f(n 1) for j = 2, 3, ..., n ==> f(n)  xj  f(n 1) ==> f(n)  f(n 1)  xj  0 marginal contribution (marginal product) again Q?: Two (i and j) deviate from the N and their share is xj and xj

Unionized workers in the landowner-worker game Union: n-1 workers refuse to work if fewer than n-1 workers are hired flat wage rate: w  0 (= xi , for i = 2, 3,...n) implicit assumption: identical workers and decreasing marginal product The core marginal product of the n-th worker f(n)-f(n-1)  w  0 if f(n)-f(n-1)<w, the owner will not hire the n-th worker so that he will not hire any one so that the owner gets f(1)>0 if and only if x1  f(1) and f(n)-f(n-1)  w  0 the grand coalition is stable A competitive equilibrium in labor market Is there a free-rider problem?

Graphical Representation of the landowner-worker game In a free labor market n* is the optimal number of employed workers w* is the competitive wage rate Stability of the core a deviation from n* i > n* ==> f’(i)<f’(n*)=w* i < n* ==> f’(i)>f’(n*)=w* a change in labor supply or VMP (=P*MP) ==> n* changes But in a unionized workers?

Graphical Representation of the Union In a unionized workers 協商容易破裂 當生產力 (MP) 或勞動供給的變動 勞資雙方對合理的 w0 認知不同 Free rider 自由市場下之 core 空間比較大 Q: 勞工會不會被剝削? Alternative job choice Experiments in the Ultimatum Game

The Shapley Value A reasonable or “fair” way to divide the gains from cooperation In a 2-person cooperative game with transferable payoffs don’t cooperate ==> v1=v({1}), v2=v({2}) cooperate ==> v(N)=v({1, 2}) marginal revenue (MR) from forming a grand coalition MR= v(N)  v({1})  v({2}) Shapley value: egalitarian solution (五五分帳) under equal contribution Shi = v({i}) + 1/2 MR

The Shapley Value

Shapley Value under different marginal contribution A Glove Game (手套配對) 3 player {1, 2, 3} player 1, 2 只有左手套, player 3 有右手套 Assumption: matched glove produce 1 unit output Payoffs v(N)=v({1, 2, 3})=1 v({1, 3})= v({2, 3})=1, and v({1, 2})=0 v({1})=v({2}) =v({3})=0 marginal contribution of player 3 (依加入N的次序) {1,2,3} ==> 1 {1,3,2} ==> 1 {2,1,3} ==> 1 {2,3,1} ==> 1 {3,1,2} ==> 0 {3,2,1} ==> 0 Average = 4/6 = 2/3 so Sh3 = 2/3

Shapley value for player 1? marginal contribution of player 1 (依加入N的次序) {1,2,3} ==> 0 {1,3,2} ==> 0 {2,1,3} ==> 0 {2,3,1} ==> 0 {3,1,2} ==> 1 {3,2,1} ==> 0 Sh1 = 1/6 A general case : the ordering of joining N S(, i) the set of players that come before i in the ordering  S(, i)  N marginal contribution from i’s joining: m(S(, i), i) average marginal contribution

An application 你擁有一個專利技術, 可能的市場(淨)價值 = 20000 (百萬元) 你願意找人合作 (入股) 嗎? Why? 假設你自有資金十分充足, 你還願意找人合作 (合夥)嗎?

Cooperative game and Risk 由於不確定所造成的衝擊 由機率來描述 payoff 的不確定: R~N(, 2) Risk averse 風險趨避 若 risk 對 DM 而言是負面的影響 (A) E(R)=  = 100, (B) RS = 100 (for sure) if DM prefers (B) to (A) , then DM is a risk averser Willing to Pay to avoid the risk? E(R)=100 (50% 率得到0, 但50%機率得200) 你願意用多少錢來換掉此不確定性? 假設 DM 願意少拿40來換取確定得60的機會 Certianty equivalent CE: 100-g()=60 g(): disutility of risk 每個人可能會不相同

Certianty Equivalent function Utility function with risk consideration If a risky payoff R~N(, 2), the general form of CE fucntion for i: CE = -(0.5/ri) 2 r: risk tolerance coefficient (風險忍受係數, r愈大愈能忍受風險) Numerical example If a DM’s r = 20000, a risky payoff= R~(35000, 250002) CE = 35000-(0.5/20000)x 250002 = 35000-15625 = 19375

合夥與釋股 DM1擁有一專利技術 (r1 = 20000) 預期價值 =35000, 標準差 =25000 預期價值 =35000, 標準差 =25000 可能的合夥人 DM2 (r2 = 30000) Two-player partnership game Payoffs v({1}) = CE= -(0.5/r1) 2, v({2}) = 0 grand coalition with equal shares: v(N) = CE1 + CE2 CE1 = (1/2) -(0.5/r1) ((1/2))2 CE2 = (1/2) -(0.5/r2) ((1/2))2 The core is empty or nonempty?

DM1 釋股50%給DM2 DM1 獨力持有 100%股權 DM1、DM2 各持有 50% 股權 CE = 35000-(0.5/20000)x 250002 = 35000-15625 = 19375 DM1、DM2 各持有 50% 股權 CE1 = (1/2) -(0.5/r1) ((1/2))2 = 17500-(0.5/20000)*(12500^2) = 13594 CE2 = (1/2) -(0.5/r2) ((1/2))2 = 17500-(0.5/35000)*(12500^2) = 14896 CE < CE1 + CE2 =28490 <== v(N)  kK v(Sk) Cohesive coalitional game DM2’s max. WTP for 50% shares =? = 14896 DM2’s min. WTA (accept) for 50% shares =? = 19375 - 13594=5781 WTP>WTA ==> non-empty core

DM1 最佳釋股比例? DM1 應該釋出多少比例的股權給 DM2? Two-player partnership game Payoffs v({1}) = CE= -(0.5/r1) 2, v({2}) = 0 grand coalition with selling  % shares to DM2: v(N) = CE1 + CE2 CE1 = (1-) -(0.5/r1) ((1- ))2 CE2 = () -(0.5/r2) (())2 Your Homework: (1) calculate  =? (2) demonstrate the core is non-empty (3) if DM2’s r2 = 10000, analyze the core