Information, Control and Games

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 improvementIncreasing 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 • 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 • Unanimity: 一致同意 • Scenario • 2 players • action : Yes/No forming a grand coalition if both say “Yes” => N={1,2}otherwise Si = {i}, i=1,2 • payoffsutilityv1 = v2 = 1 if 2 players are in the grand coalitionvi (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}} • payoffsv({i}) = 0 for i = 1,2,3v(S) = 1 for every other coalition

Landowner and Workers game • 地主可自行耕作, 或雇用n個工人 • 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 payoffA coalitional game has transferable payoffif 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 payoffv(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}) v({1,3})

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

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 Payoff space in a simplex form

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} • V({i}) = 0 for i = 1, 2

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 gamev(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 constraintsx1  p and x2  q • Additional constraints in graphical representation

3-player cooperative game and the core • Three-player majority game • Payoffsv({i}) = 0 for i = 1,2,3v(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 ruleAny one in grand coalition shares 1/3Any 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 • Payoffsv({i}) = 0 for i = 1,2,3two as the majority: v(M) =0grand 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 = 1,2,3two as the majority: v(M) = grand coalition: v(N)=1 •  = ?

M={1,3} (v1, v3)| iM = (/2,  /2) N={1,2,3} (v1, v2, v3)| iN = (1/2, 0, 1/2) 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) =(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 • v(N)=1, vi = 1/3 for i = 1, 2, 3 • v(M)= (此例  = 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

Empty and Non-Empty Core in a simplex form Empty core Nonempty core

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 ifx1 + 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 payoffsx1 + 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 alonex1 f(1), (4) x2 0, andx3 0

Another Aspect • Marginal contribution • x1 + x2 + x3 = f(3) ==> x1 = f(3) - x2 - x3substitute 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: ngrand 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 workerf(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 oneso that the owner gets f(1)>0 • if and only if x1 f(1) and f(n)-f(n-1)  w  0the 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 coalitionMR= 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 • Payoffsv(N)=v({1, 2, 3})=1v({1, 3})= v({2, 3})=1, and v({1, 2})=0v({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 • 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()=60g(): 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 • 可能的合夥人 DM2 (r2 = 30000) • Two-player partnership game • Payoffsv({1}) = CE= -(0.5/r1) 2,v({2}) = 0grand coalition with equal shares: v(N) = CE1 + CE2CE1 = (1/2) -(0.5/r1) ((1/2))2CE2 = (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 • 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? • 賣給 DM2 比例 • Two-player partnership game • Payoffsv({1}) = CE= -(0.5/r1) 2,v({2}) = 0grand coalition with selling  % shares to DM2: v(N) = CE1 + CE2CE1 = (1-) -(0.5/r1) ((1- ))2CE2 = () -(0.5/r2) (())2 • Your Homework: • (1) calculate  =? • (2) demonstrate the core is non-empty • (3) if DM2’s r2 = 10000, analyze the core

Information, Control and Games