Game Theory

1. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 1 Game Theory By Chattrakul Sombattheera Supervisors A/Prof Peter Hyland & Prof Aditya Ghose

2. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 2 Game theory Analysis of problems of conflict and cooperation among independent decision-makers. Players, having partial control over outcomes of the game, are eager to finish the game with an outcome that gives them maximal payoffs possible Emile Borel, a French mathematician, published several papers on the theory of games in 1921 Von Neumann & Morgenstern�s The Game Theory and Economics Behavior in 1944 A convenient way in which to model the strategic interaction problems eg. Economics, Politics, Biology, etc.

3. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 3 The Games Game = <Rules, Components> Rules: descriptions for playing game Components: A set of rational players A set of all strategies of all players A set of the payoff (utility) functions for each combination of players� strategies A set of outcomes of the game A set of Information elements

4. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 4 Modeling Game The rules give details how the game is played e.g. How many players, What they can do, and What they will achieve, etc. Modeler study the game to find equilibrium, a steady state of the game where players select their best possible strategies. To find equilibrium = to find solution = to solve games

5. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 5 Player and Rationality Player can be a person, a team, an organization In its mildest form, rationality implies that every player is motivated by maximizing his own payoff. In a stricter sense, it implies that every player always maximizes his payoff, thus being able to perfectly calculate the probabilistic result of every strategy.

6. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 6 Movement of the Game Simultaneous: All players make decisions (or select a strategy) without knowledge of the strategies that are being chosen by other players. Sequential: All players make decisions (or select a strategy) following a certain predefined order, and in which at least some players can observe the moves of players who preceded them Games can be played repeatedly

7. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 7 Information Information is what the players know while playing games: All possible outcomes The payoff/utility over outcomes Strategies or actions used An item of information in a game is common knowledge if all of the players know it and all of the players know that all other players know it

8. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 8 Information Complete information: if the payoffs of each player are common knowledge among all the players Incomplete information: if the payoffs of each player, or certain parameters to it, remain private information of each player. Perfect Information: Each player knows every strategy of the players that moved before him at every point. Imperfect Information: If a player does not know exactly what strategies other players took up to a point.

9. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 9 Strategies A strategy is a comprehensive plan of actions � what actions to be played based on information available Each player has a set of strategies In a simple form, a strategy merely dictates players to perform an action

10. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 10 Strategies Player I, SI = {x1, x2} Player II, SII = {y1, y2, y3} Player III, SIII = {z1, z2} S is a set of 12 combinations of strategies Each combination of strategy is an action (strategy) profile e.g. (x1, y2, z1)

11. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 11 Outcome, Utility In general, outcome is a set of interesting elements that the modeler picks from the value of actions, payoffs, and other variables after the game terminates. Outcomes are often represented by action (strategy) profiles Utility represents the motivations of players. A utility function for a given player assigns a number for every possible outcome of the game with the property that a higher number implies that the outcome is more preferred. Utility functions may either ordinal in which case only the relative rankings are important, but no quantity is actually being measured, or cardinal, which are important for games involving mixed strategies

12. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 12 Payoff Payoffs are numbers which represent the motivations of players. Payoffs may represent profit, quantity, "utility," or other continuous measures (cardinal payoffs), or may simply rank the desirability of outcomes (ordinal payoffs). In most of this presentation, we assume that utility function assigns payoffs

13. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 13 Variety of Game Game can be modelled with variety of its components We introduce Non-cooperative form game Normal (strategic) form game Extensive form game Cooperative form game Characteristic function game

14. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 14 Normal (Strategic) Form Game An n-person game in normal (strategic) form is characterised by A set of players N = {1, 2, 3, �, n} A set S = S1 x S2 x � x Sn is the set of combinations of strategy profiles of n players Utility function ui : S ? R of each player

15. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 15 Normal (Strategic) Form Game Components of a normal form game can be represented in game matrix or payoff matrix Game matrix of 2 players: Player I and Player II Each player has a finite number of strategies S1 = {s11, s12} S2={s21, s22}

16. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 16 Zero Sum Game Von Neumann and Morgenstern studied two-person games which result in zero sum: one player wins what the other player loses The payoff of player II is the negative value of the payoff of player I =

17. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 17 Matching Pennies Player I & Player II: Choose H or T (not knowing each other�s choice) If coins are alike, Player II wins $1 from Player I If coins are different, Player I wins $1 from Player II

18. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 18 Pure Strategy A prescription of decision for each possible situation is known as pure strategy A pure strategy can be as simple as : Play Head, Play Tail A pure strategy can be more complicated as : Play Head after wining a game We refer to each of strategies of a player as a pure strategy

19. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 19 Maximax Strategy �Maximax principle counsels the player to choose the strategy that yields the best of the best possible outcomes.� Two players simultaneously put either a blue or a red card on the table If player I puts a red card down on the table, whichever card player II puts down, no one wins anything If player I puts a blue card on the table and player II puts a red card, then player II wins $1,000 from player I Finally, if player I puts a blue card on the table and player II puts a blue card down, then player I wins $1,000 from player II

20. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 20 Maximax Strategy With maximax principle, player I will always play the blue card, since it has the maximum possible payoff of 1,000. Player II is rational, he will never play the blue card, since the red card gives him 1,000 payoff. In such a case, if player I plays by the maximax rule, he will always lose. The maximax principle is inherently irrational, as it does not take into account the other player's possible choices. Maximax is often adopted by naive decision-makers such as young children.

21. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 21 Battle of the Pacific In 1943, the Allied forces received reports that a Japanese convoy would be heading by sea to reinforce their troops. The convoy could take on of two routes -- the Northern or the Southern route. The Allies had to decide where to disperse their reconnaissance aircraft -- in the north or the south -- in order to spot the convoy as early as possible. The payoff matrix shows payoffs expressed in the number of days of bombing the Allies could achieve

22. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 22 Minimax Strategy Minimax strategy is to minimize the maximum possible loss which can result from any outcome. To cause maximum loss to the Japanese, the Allies would like to go South To avoid maximum loss, in case the Allies go South, the Japanese would go North If the Japanese go North, the Allies would go North to maximize their payoff

23. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 23 Domination in Pure Strategy Player I selects a row while Player II selects a column in response to each other for their maximum payoffs Player II�s F strategy is always better than G no matter what strategy Player I selects Strategy G is dominated by F, or F is a dominant strategy rational player never plays dominated strategies.

24. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 24 Solving Pure Strategy Player I selects a row while Player II selects a column in response to each other for their maximum payoffs Player I selects D for maximum payoff (16), Player II selects E for his maximum payoff (-16) Player I then selects A, while Player II selects F Player I selects C, while Player II cannot improve

25. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 25 Pure Strategy: Saddle Point Strategies (C,F) is an equilibrium outcome, players have no incentives to leave At (C,F), I knows that he can win at least 2 while II knows that he can lose at most 2 The value 2 at (C,F) is the minimum of its row and is the maximum of its columns� it is call the Saddle point or the value of the game The saddle point is the game�s equilibrium outcome A game may have a number of saddle points of the same value

26. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 26 Mixed-Strategies: Odd or Even A player can randomly take multiple actions (or strategies) based on probability� mixed strategies Player I and Player II simultaneously call out one of the numbers one or two. Player I wins if the sum of the number is odd Player II win if the sum of the number is even Note: Payoffs in dollars.

27. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 27 Solving Odd or Even Suppose Player I calls �one� 3/5ths of the times and �two� 2/5ths of the times at random If II calls �one�, I loses 2 dollars 3/5ths of the times and wins 3 dollars 2/5ths of the times. On average, I wins -2(3/5) + 3(2/5) = 0 If II calls �two�, I wins 3 dollars 3/5ths of the times and loses 4 dollars 2/5ths of the times, On average, I wins -3(3/5) � 4(2/5) = 1/5

28. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 28 Solving Odd or Even I win 0.20 on average every time II calls �two� Can I fix this so that he wins no matter what II plays?

29. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 29 Equalizing Strategy Let p be a probability Player I calls �one� such that I wins the same amount on average no matter what II calls Since I�s average winnings when II calls �one� and �two� are -2p+3(1-p) and 3p-4(1-p), respectively. So� -2p + 3(1-p) = 3p-4(1-p) 3 � 5p = 7p � 4 12p = 7 p = 7/12

30. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 30 Equalizing Strategy Therefore, I should call �one� with probability 7/12 and two with 5/12 On average, I wins -2(7/12) + 3(5/12) = 1/12 or 0.0833 every play regardless of what II does. Such strategy that produces the same average winnings no matter what the opponent does is called an equalizing strategy

31. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 31 Minimax Strategy In Odd or Even, Player I cannot do better than 0.0833 if Player II plays properly Following the same procedure, II calls �one� with probability 7/12 �two� with probability 5/12 If I calls �one�, II�s average loss is -2(7/12) + 3(5/12) = 1/12 If I calls �two�, II�s average loss is 3(7/12) � 4(5/12) = 1/12 1/12 is called the value of the game or the saddle point Mixed strategies used to ensure this are called optimal strategy or minimax strategy

32. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 32 Minimax Theorem A two person zero sum game is finite if both strategy set Si and Sj are finite sets. For every finite two-person zero-sum game There is a number V, call the value of the game There is a mixed strategy for Player I such that I�s average gain is at least V no matter what II does, and There is a mixed strategy for Player II such that II�s average loss is at most V no matter what I does

33. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 33 Non-Zero Sum Game The sum of the utility is not zero Prisoner Dilemma Nash equilibrium Chicken Stag Hunt

34. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 34 Prisoner Dilemma Two suspects in a crime are held in separate cells There is enough evidence to convict each one of them for a minor offence, not for a major crime One of them has to be a witness against the other (finks) for convicting major crime If both stay quiet, each will be jailed for 1 year If one and only one finks , he will be freed while the other will be jailed for 4 years If both fink, they will be jailed for 3 years

35. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 35 Prisoner Dilemma Utility function assigned u1(F,Q) = 4, u1(Q,Q) = 3, u1(F,F) = 1, u1(Q,F) = 0 u2(Q,F) = 4, u2(Q,Q) = 3, u2(F,F) = 1, u2(F,Q) = 0 What should be the outcome of the game? Both would prefer Q But they have incentive for being freed, choose F

36. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 36 Prisoner Dilemma Prisoner I: Acting Fink against Prisoner II�s Quiet yields better payoff than Quiet. Fink is called the best strategy against Prisoner II�s Quiet Prisoner I: Acting Fink against Prisoner II�s Fink yields better payoff than Quiet. Fink is the best strategy against Prisoner II�s Fink

37. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 37 Dominant Strategy A dominant strategy is the one that is the best against every other player�s strategy. Prisoner I: Fink is the dominant strategy Prisoner II: Fink is the dominant strategy Outcome (1,1) is called dominant strategy equilibrium

38. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 38 Nash Equilibrium John Nash, the economics Nobel Winner. An action (strategy) profile a = (a1, a2, a3, �, an) is combination of action ai, selected from player i strategy Si Nash equilibrium is �an action profile a* with the property that no player i can do better by choosing an action different from ai*, given that every other player j adheres to aj*.�

39. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 39 Nash Equilibrium & Strategies Nash equilibrium is �an action profile a* with the property that no player i can do better by choosing an action different from ai*, given that every other player j adheres to aj*.� Players = {I, II, III} SI={x1, x2}, SII={y1, y2, y3}, SIII={z1, z2} (x1, y2, z1) is a Nash Equilibrium if uI(x1, y2, z1) = uI (x2, y2, z1) and uII(x1, y2, z1) = uII (x1, y1, z1) and uII(x1, y2, z1) = uII (x1, y3, z1) and uIII(x1, y2, z1) = uIII (x1, y2, z2)

40. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 40 Nash Equilibrium What is the equilibrium in Prisoner Dilemma? Usually, dominant equilibrium is Nash equilibrium But, Nash Equilibrium may not be dominant equilibrium

41. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 41 Stag Hunt Game Each of a group of hunters has two options: he may remain attentive to the pursuit of a stag, or catch a hare If all hunters pursue the stag, they catch it and share it equally If any hunter devotes his energy to catching a hare, the stag escape, and the hare belongs to the defecting hunter alone Each hunter prefers a share of the stag to a hare

42. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 42 Stag Hunt & Equilibrium A group of 2 hunters value payoffs are u1(stag, stag) = u2(stag, stag) = 2, u1(stage,hare) = 0, u2(stage,hare) = 1, u1(hare,stag) = 1, u2(hare,stag) = 0 and u1(hare,hare) = u2(hare,hare) = 1 There are 2 equilibria (stag, stag) and (hare, hare)

43. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 43 Chicken There are two hot �Gong teenagers, Smith and Brown Smith drives a V8 Commodore heading South down the middle of Princes Hwy, and Brown drives V8 Falcon up North When approaching each other, each has the choice to stay in the middle or swerve The one who swerves is called �chicken� and loses face, the other claims brave-hearted pride If both do not swerve, they are killed But if they swerve, they are embarrassingly called chicken

44. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 44 Chicken & Nash Equilibrium The cardinal payoffs are u(stay, stay) = (-3,-3), u(stay, swerve) = (2,0), u(swerve, stay) = (0,2) and u(swerve, swerve) = (1,1) There is no dominant strategy but there are two pure strategy Nash equilibria (swerve, stay) and (stay, swerve) How do the players know which equilibrium will be played out?

45. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 45 Chicken In mixed strategies, both must be indifferent between swerve and stay Let p be the probability for Brown to stay -3p = 2p + 1(1-p) p = 1/4 = 0.25 The chance for being survival is 1 � (p * p) 1 � 0.0625 = 0.9375

46. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 46 Game with No Equilibrium Matching Pennies: Player 1 & Player 2 choose H or T (not knowing each other�s choice) If coins are alike, Player 2 wins $1 from Player 1 If coins are different, Player 1 wins $1 from Player 2 There is no Nash equilibrium pure strategy There, however, is a Nash equilibrium mixed strategy where each player plays head with probability 0.5 The average payoffs for both players are 0

47. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 47 Nash Equilibrium In equilibrium, each player is playing the strategy that is a "best response" to the strategies of the other players. No one has an incentive to change his strategy given the strategy choices of the others Game may not have equilibrium Game may have equilibria Equilibrium is not the best possible outcome !!!

48. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 48 Pareto Optimum Named after Vilfredo Pareto, Pareto optimality is a measure of efficiency An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player better off A Pareto Optimal outcome cannot be improved upon without hurting at least one player. Often, a Nash Equilibrium is not Pareto Optimal implying that the players' payoffs can all be increased.

49. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 49 Equilibrium and Optimum In Prisoner Dilemma, both players have incentives to leave {Fink, Fink} One will earn more but the other will be worst off. {Q, Q} is Pareto optimal Nash equilibrium does not guarantee optimality

50. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 50 Equilibrium & Optimum In Stag Hunt, there are 2 equilibria (stag, stag) and (hare, hare) Only one of the equilibria is optimal

51. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 51 Equilibrium & Optimum In Chicken game, equilibria are (Swerve, Stay) and (Stay, Swerve) Both of equilibria have one swerve and one stay Both equilibria are Pareto optimal

52. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 52 Extensive Form Game Game in extensive form can be represented by a topological tree or game tree A topological tree is a collection of finite nodes Each node is connected by a link There is a unique sequence of nodes and links between any pair of nodes Node C follows B if the sequence of links joining A to C passes through B Node X is called terminal if no nodes follows X

53. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 53 Extensive Form Game An n-person game in extensive form is characterised by A tree T, with a node A called the starting point of T A utility function, assigning an n-vector to each terminal node of T A partition of the non-terminal nodes of T into n + 1 sets, Si, called the player sets

54. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 54 Extensive Form Game A probability distribution, defined at each node of Si among the intermediate followers of this vertex. For each player i, there is a sub-partition of Si into subsets Sij called information set For each information set Sij, All nodes have the same number of outgoing links Every path from root to terminal nodes can cross Sij only once

55. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 55 Extensive Form Game

56. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 56 Parlor Game Parlor game is an extensive form game The rule specify a series of well-defined moves A move is a point of decision for a given player from among a set alternatives. A particular alternative chosen by a player at a given decision point is a choice. Sequence of choices until the game is terminated is a play.

57. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 57 A Modified Spade Game For simplicity, a set of cards are reduced to aces, 2s, and 3s. A deck of cards is divided into suits, one of which (say clubs) is discarded. A second suit (spades) is shuffled and placed face down on the table Each of the two players has in his hand a complete suit The cards are valued: ace = 1, 2 = 2 and 3 = 3 The spades are turned over one by one and each is bided by one of the players, the one capturing the larger value of spades wins (46.) The first spade is turned over then the player simultaneously bid for the spade with a card in his hand: the higher value wins If a draw occurs, the winner of the next round takes the spades

58. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 58 Modified Spade Game

59. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 59 Matching Pennies in Extensive form Game Player 1 & Player 2: Choose H or T (not knowing each other�s choice) If coins are alike, Player 2 wins $1 from Player 1 If coins are different, Player 1 wins $1 from Player 2

60. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 60 X-O in Extensive Form Game

61. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 61 Cooperative Game Players can communicate (negotiate) Players can make binding agreement (forming coalition) Players can make side payment (transferable utility)

62. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 62 Coalitions in Cooperative Game N is a set of players A coalition S is a subset of N, a set of all coalitions is denoted by S The set N is also a coalition, specially called grand coalition A coalition structure is a set CS = {S1, S2, �, Sm} which is a partition of N such that Sj ? ?, j = 1, 2, �, m Si ? Sj = ? for all i ? j S1 ? S2 ? � ? Sm = N

63. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 63 Coalitions in Cooperative Game N = {1, 2, 3} is a set of players All possible coalitions are S ={?, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Coalition structures are {{1}, {2}, {3}}, {{1}, {2,3}}, {{1,2}, {3}}, {{1,3}, {2}} and {{1,2,3}}

64. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 64 Payoff in Cooperative Game A game eventually terminates in an end-state i.e. outcome or coalition structure. The quantitative representation of an outcome to a player is a payoff xi. A collection of payoffs to all players is a payoff vector x = (x1, x2, x3, �, xn) A payoff configuration is a pair of a payoff vector and a coalition structure denoted by (x; CS) = {x1, x2, x3,�, xn; S1, S2, �, Sm}

65. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 65 Cooperative Game in Characteristic Function Form An n-person game in characteristic function form is characterized by a pair (N:?) where N = {1, 2, �, n} is a set of players; n = 2 v : S ? R is a characteristic function defining a real value to each coalition S of N. Thus the game is named Characteristic Function Game (CFG)

66. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 66 Characteristic Function Game Implicit assumptions: The value of any coalition is in money, and the players prefer more money to less A coalition S forms by making a binding agreement on the way its value v(S) is to be distributed among its members. The amount v(S) does not in anyway depend on the actions of N-S, though N-S might partition it self into coalitions. The amount v(S) cannot given to any player outside S.

67. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 67 Characteristic Function Game The characteristic function v is known to all players. Any agreement concerning the formation and disbursement of value is known to all n players as soon as it is made. The characteristic function influences player affinities for each other. Every nonempty coalition can form.

68. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 68 Characteristic Function Game Odd Man Out, an example of CFG: Three players {1, 2, 3} bargain in pairs to form a deal, dividing money, depending on coalitions If 1 and 2 combine, excluding 3, they split $4.0 If 1 and 3 combine, excluding 2, they split $5.0 If 2 and 3 combine, excluding 1, they split $6.0

69. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 69 Characteristic Function Game Odd Man Out�s characteristic function: v({1,2})=4, v ({1,3})=5, v ({2,3})=6 v({1}) = v({2}) = v({3}) = v({1,2,3}) = 0 Possible payoff configurations (2.0, 2.0, 0: {1,2},{3}) (2.5, 0, 2.5: {1, 3}, {2}) (0, 3.0, 3.0: {1}, {2,3}) (0, 0, 0: {1}, {2}, {3})

70. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 70 Characteristic Function Game Sandal makers: Maker 1 and 2 make only left sandals, each at rate 17 pieces at a time Maker 3, 4 and 5 make only right sandals, each at rate 10 pieces at a time Any single sandal worth nothing while a pair (of left and right!) sells $20. A coalition is a binding agreement between left and right sandal makers.

71. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 71 Characteristic Function Game Sandal makers characteristic function: v({1}) = v({2}) = v({3}) = v({4}) = v({5}) = 0 v({1,2}) = v({3,4}) = v({3,5}) = v({4,5}) = v({3,4,5}) = 0 v({1,3}) = v({2,3}) = v({1,4}) = v({2,4}) = v({1,5}) = v({2,5}) = 200 v({1,2,3}) = v({1,2,4}) = v({1,2,5}) = 200 v({1,3,4}) = v({1,3,5}) = v({1,4,5}) = v({2,3,4}) = v({2,3,5}) = v({2,4,5}) = 340 v({1,3,4,5}) = v({2,3,4,5}) = 340 v({1,2,3,4}) = v({1,2,3,5}) = v({1,2,4,5}) = 400 v({1,2,3,4,5}) = 600

72. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 72 Characteristic Function Game (CFG) Possible payoff configurations: (100, 100, 100, 100, 0: {1,3}, {2,4}, {5}) (100, 100, 100, 100, 0: {1,4}, {2,3}, {5}) (113.3, 100, 113.3, 113.3, 100: {1,3,4}, {2,5}) (100, 100, 100, 100, 0:{1,2,3,4},{5}) (120, 120, 120, 120, 120: {1,2,3,4,5})

73. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 73 Super-Additive Environment A game is super-additive if v(S ? T) ? v(S) + v(T) for all S, T ? N such that S ? T = �. In a super-additive environment, e.g. sandal makers or social welfare, players tend to form a grand coalition. In a non-super-additive environment, self-interested players make the game very interesting.

74. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 74 Stability and Efficiency in CFG The game is played and is meant to reach a stable state: no player has incentive to leave coalition or change strategy Core. The issue of how well/fair the payoffs are distributed is efficiency Shapley value .

75. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 75 Imputation in Super-additive Von Neumann & Morgenstern believed the distribution of coalition values is the key to successful coalition formation. Let the value of a singleton coalition of player i v(i) is denoted by vi, payoff vectors should hold Individual rationality: xi ? vi for all i Collective rationality: ? xi = v(N) An imputation is a payoff vector x = (x1, x2, � xn) satisfying xi ? vi and ? xi = v(N) for all i

76. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 76 Imputation in Sandal Makers (100, 100, 100, 100, 0: {1,3}, {2,4}, {5}) (100, 100, 100, 100, 0: {1,4}, {2,3}, {5}) (113.3, 100, 113.3, 113.3, 100: {1,3,4}, {2,5}) (100, 100, 100, 100, 0:{1,2,3,4},{5}) (120, 120, 120, 120, 120: {1,2,3,4,5})

77. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 77 Modified Sandal Makers All maker make 15 sandals, characteristic functions are v({1}) = v({2}) = v({3}) = v({4}) = v({5}) = 0 v({1,2}) = v({3,4}) = v({3,5}) = v({4,5}) = v({3,4,5}) = 0 v({1,3}) = v({2,3}) = v({1,4}) = v({2,4}) = v({1,5}) = v({2,5}) = 300 v({1,2,3}) = v({1,2,4}) = v({1,2,5}) = 300 v({1,3,4}) = v({1,3,5}) = v({1,4,5}) = v({2,3,4}) = v({2,3,5}) = v({2,4,5}) = 300 v({1,3,4,5}) = v({2,3,4,5}) = 300 v({1,2,3,4}) = v({1,2,3,5}) = v({1,2,4,5}) = 600 v({1,2,3,4,5}) = 600

78. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 78 Imputation of Modified Sandal Makers Payoff configuration (150, 150, 150, 150, 0: {1,2,3,4}, {5}) (150, 150, 150, 0, 150: {1,2,3,5}, {4}) (150, 150, 0, 150, 150: {1,2,4,5}, {3}) (120, 120, 120, 120, 120: {1,2,3,4,5})

79. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 79 Imputation, Domination and the Core Imputation x dominates y over S ? N if xi > yi for all i in S and ? xi ? v(S) The core is the set of all undominated imputations in the game Only imputations in the core can persist in negotiations The core can be empty

80. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 80 The Core of Sandal Makers (100, 100, 100, 100, 0: {1,3}, {2,4}, {5}) (100, 100, 100, 100, 0: {1,4}, {2,3}, {5}) (113.3, 100, 113.3, 113.3, 100: {1,3,4}, {2,5}) (100, 100, 100, 100, 0:{1,2,3,4},{5}) (120, 120, 120, 120, 120: {1,2,3,4,5})

81. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 81 Core of Modified Sandal Makers Payoff configuration (150, 150, 150, 150, 0: {1,2,3,4}, {5}) (150, 150, 150, 0, 150: {1,2,3,5}, {4}) (150, 150, 0, 150, 150: {1,2,4,5}, {3}) (120, 120, 120, 120, 120: {1,2,3,4,5}) (150, 150, 100, 100, 100: {1,2,3,4,5})

82. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 82 Shapley Value In a game (N,v), Shapley proposed a concept of fair distribution of payoff ? = (?1, ?2, ..., ?n), which is captured in three axioms. I: ? should only depend on v, if players i and j have symmetric roles then ?i = ?j II: If v(S) = v(S - i) for all coalition S ? N, then ?i = 0. Adding a dummy player i does not change the value ?j for other players j in the game III: If (N, v) and (N, w) are two different games, and the sum game v + w is defined as (v + w)(S) = v (S) + w (S) for all coalitions S, then ?[v+w] = ? [v] + ?[w]

83. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 83 Shapley Value To calculate the payoff, consider the players forming grand coalition step by step Start by one player and add each additional player As each player joins, award the new player an additional value he contributes to the coalition Once this is done for each of the n! grand coalitions divide the accumulated awards to each player by n! to give the fair imputation ?

84. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 84 Shapley Value Consider the following game. v(A) = v(B) = v(C) = 0 v(AB) = 2, v(AC) = 4, v(BC) = 6 v(ABC) = 7 The 6 (3!) ordered grand coalitions are:

85. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 85 Shapley Value Considering the ordered BCA coalition, the value added by each player is: B: v(B) - v(?) = 0 - 0 = 0 C: v(BC) - v(B) = 6 - 0 = 6 A: v(BCA) - v(BC) = 7 - 6 = 1 v(A) = v(B) = v(C) = 0 v(AB) = 2, v(AC) = 4, v(BC) = 6 v(ABC) = 7 ? = 1/6(8, 14, 20) = (1.33, 2.33, 3.33)

86. 6/08/2012 Coalition Formation Roadmap: Chattrakul Sombattheera 86 Done ! Questions � Comments�?