1. Information, Control and Games
Shi-Chung Chang
EE-II 245, Tel: ext. 245
Office Hours: Mon/Wed 1:00-2:00 pm or by appointment
Yi-Nung Yang
(03 ) ext. 5205,

2. Introduction to Cooperative Games
Coalitional games

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. Non-linear transferable payoff
Payoff possibility set

11. 3-player payoff space
Non-transferable payoffs

12. 3-player payoff space Transferable payoffs v(N) v({2,3}) v({1,2})

13. 3-player payoff space (non-linear) Transferable payoffs v(N) v({2,3})

14. 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

15. 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}

16. 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

17. 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

18. 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

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

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

21. 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

22. 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 / = 0 otherwise
The core is Empty?

23. 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
 = ?

24. 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)

25. 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)

26. 3-player cooperative game in the simplex form
Majority rule with equal shares
同樣的想法類推...

27. 3-player cooperative game and the graphics
The largest  with non-empty core occurs in the interception of the tree blue lines

28. 3-player cooperative game and the graphics
The largest  with non-empty core occurs in the interception of the tree blue lines

29. Empty and Non-Empty Core in a simplex form

30. 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)

31. The core of voting trading game

32. 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

33. 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

34. 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

35. 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 + x 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 + x 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

36. 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?

37. 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?

38. 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

39. 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

40. The Shapley Value

41. 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

42. 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

43. 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?

44. 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

45. 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, ) CE = (0.5/20000)x = = 19375

46. 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?

47. DM1 釋股50%給DM2 DM1 獨力持有 100%股權 DM1、DM2 各持有 50% 股權
CE = (0.5/20000)x = = 19375
DM1、DM2 各持有 50% 股權
CE1 = (1/2) -(0.5/r1) ((1/2)) = (0.5/20000)*(12500^2) = 13594
CE2 = (1/2) -(0.5/r2) ((1/2)) = (0.5/35000)*(12500^2) = 14896
CE < CE1 + CE2 = <== 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 =?
= =5781
WTP>WTA ==> non-empty core

48. 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