1 MATH 1020 Chapter 1: Introduction to Game theory Dr. Tsang

Why do we like games? amusement, thrill and the hope to win uncertainty – course and result of a game 2

Reasons for uncertainty randomness combinatorial multiplicity imperfect information 3

Three types of games 4 bridge

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Game Theory 博弈论 Game theory is the study of how people interact and make decisions. This broad definition applies to most of the social sciences, but game theory applies mathematical models to this interaction under the assumption that each person's behavior impacts the well-being of all other participants in the game. These models are often quite simplified abstractions of real-world interactions. 6

A cultural comment The Chinese translation “ 博弈论 ” may be a little bit misleading. Games are serious stuffs in western culture. – The Great Game : the strategic rivalry and conflict between the British Empire and the Russian Empire for supremacy in Central Asia ( ). – Wargaming: informal name for military simulations, in which theories/tactics of warfare can be tested and refined without the need for actual hostilities. 7

The Great Game: 8 Political cartoon depicting the Afghan Emir Sher Ali with his "friends" the Russian Bear and British Lion (1878)

What is Game Theory? Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome “how rational individuals make decisions when they are aware that their actions affect each other and when each individual takes this into account” 9

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Brief History of Game Theory Game theoretic notions go back thousands of years ( Sun Tzu‘s writings 孙子兵法 ) E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined John von Neumann proves the minimax theorem John von Neumann & Oskar Morgenstern write "Theory of Games and Economic Behavior” John Nash describes Nash equilibrium (Nobel price 1994) 11

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Rationality Assumptions: humans are rational beings humans always seek the best alternative in a set of possible choices Why assume rationality? narrow down the range of possibilities predictability 13

Utility Theory Utility Theory based on: rationality maximization of utility – may not be a linear function of income or wealth 14 Utility is a quantification of a person's preferences with respect to certain behavior as oppose to other possible ones.

Game Theory in the Real World Economists innovated antitrust policy auctions of radio spectrum licenses for cell phone program that matches medical residents to hospitals. Computer scientists new software algorithms and routing protocols Game AI Military strategists nuclear policy and notions of strategic deterrence. Sports coaching staffs run versus pass or pitch fast balls versus sliders. Biologists what species have the greatest likelihood of extinction. 15

What are the Games in Game Theory? For Game Theory, our focus is on games where: – There are 2 or more players. – There is some choice of action where strategy matters. – The game has one or more outcomes, e.g. someone wins, someone loses. – The outcome depends on the strategies chosen by all players; there is strategic interaction. What does this rule out? – Games of pure chance, e.g. lotteries, slot machines. (Strategies don't matter). – Games without strategic interaction between players, e.g. Solitaire. 16

Game Theory Finding acceptable, if not optimal, strategies in conflict situations. An abstraction of real complex situation Assumes all human interactions can be understood and navigated by presumptions – players are interdependent – uncertainty: opponent’s actions are not entirely predictable – players take actions to maximize their gain/utilities 17

18 Types of games zero-sum or non-zero-sum [if the total payoff of the players is always 0] cooperative or non-cooperative [if players can communicate with each other] complete or incomplete information [if all the players know the same information] two-person or n-person Sequential vs. Simultaneous moves Single Play vs. Iterated

Essential Elements of a Game 1.The players how many players are there? does nature/chance play a role? 2.A complete description of what the players can do – the set of all possible actions. 3.The information that players have available when choosing their actions 4.A description of the payoff consequences for each player for every possible combination of actions chosen by all players playing the game. 5.A description of all players’ preferences over payoffs. 19

Normal Form Representation of Games A common way of representing games, especially simultaneous games, is the normal form representation, which uses a table structure called a payoff matrix to represent the available strategies (or actions) and the payoffs. 20

21 A payoff matrix: “to Ad or not to Ad” PLAYERS STRATEGIES PAYOFFS Philip Morris No AdAd Reynolds No Ad 50, 50 20, 60 Ad 60, 20 30, 30

The Prisoners' Dilemma The Prisoners' Dilemma 囚徒困境 囚徒困境 Two players, prisoners 1, 2. Each prisoner has two possible actions. – Prisoner 1: Don't Confess, Confess – Prisoner 2: Don't Confess, Confess Players choose actions simultaneously without knowing the action chosen by the other. Payoff consequences quantified in prison years. – If neither confesses, each gets 3 year – If both confess, each gets 5 years – If 1 confesses, he goes free and other gets 10 years – Prisoner 1 payoff first, followed by prisoner 2 payoff – Payoffs are negative, it is the years of loss of freedom 22

Prisoners’ Dilemma: Prisoners’ Dilemma: payoff matrix ConfessDon’t Confess Confess-5, -50, -10 Don’t Confess -10, 0-3,

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Prisoner’s Dilemma : Example of Non-Zero Sum Game A zero-sum game is one in which the players' interests are in direct conflict, e.g. in football, one team wins and the other loses; payoffs sum to zero. A game is non-zero-sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain. For example, when both players choose Don't Confess in the Prisoners' Dilemma 25

Zero-Sum Games The sum of the payoffs remains constant during the course of the game. Two sides in conflict Being well informed always helps a player 26

Non-zero Sum Game The sum of payoffs is not constant during the course of game play. Some nonzero-sum games are positive sum and some are negative sum Players may co-operate or compete. 27

Information Players have perfect information if they know exactly what has happened every time a decision needs to be made, e.g. in Chess. Otherwise, the game is one of imperfect information. 28

Imperfect Information Partial or no information concerning the opponent is given in advance to the player’s decision, e.g. Prisoner’s Dilemma. Imperfect information may be diminished over time if the same game with the same opponent is to be repeated. 29

Games of Perfect Information The information concerning an opponent’s move is well known in advance, e.g. chess. All sequential move games are of this type. 30

Games of Co-operation Players may improve payoff through communicating forming binding coalitions & agreements do not apply to zero-sum games Prisoner’s Dilemma with Cooperation 31

Games of Conflict Two sides competing against each other Usually caused by complete lack of information about the opponent or the game Characteristic of zero-sum games 32

Example of zero-sum game 33 Matching Pennies matcherMis-matcher

34 Rock-Paper-Scissors

35 Zero-sum game matrices are sometimes expressed with only one number in each box, in which case each entry is interpreted as a gain for row-player and a loss for column- player.

Strategies A strategy is a “complete plan of action” that fully determines the player's behavior, a decision rule or set of instructions about which actions a player should take following all possible histories up to that stage. The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game. 36

Dominant or dominated strategy A strategy S for a player A is dominant if it is always the best strategies for player A no matter what strategies other players will take. A strategy S for a player A is dominated if it is always one of the worst strategies for player A no matter what strategies other players will take. 37

If you have a dominant strategy, use it! Use strategy 1 38

39 Dominance Solvable If each player has a dominant strategy, the game is dominance solvable COMMANDMENT If you have a dominant strategy, use it. Expect your opponent to use his/her dominant strategy if he/she has one.

40 Only one player has a Dominant Strategy For The Economist: – G dominant, S dominated Dominated Strategy: There exists another strategy which always does better regardless of opponents’ actions The Economist GS Time S 100, 100 0, 90 G 95, , 90

How to recognize a Dominant Strategy 41 To determine if the row player has any dominant strategy 1.Underline the maximum payoff in each column 2.If the underlined numbers all appear in a row, then it is the dominant strategy for the row player No dominant strategy for the row player in this example.

How to recognize a Dominant Strategy 42 To determine if the column player has any dominant strategy 1.Underline the maximum payoff in each row 2.If the underlined numbers all appear in a column, then it is the dominant strategy for the column player There is a dominant strategy for the column player in this example.

43 If there is no dominant strategy Does any player have a dominant strategy? If there is none, ask “Does any player have a dominated strategy?” If yes, then Eliminate the dominated strategies Reduce the normal-form game Iterate the above procedure

Eliminate any dominated strategy Eliminate strategy 2 as it’s dominated by strategy 1 44

45 Successive Elimination of Dominated Strategies If a strategy is dominated, eliminate it The size and complexity of the game is reduced Eliminate any dominant strategies from the reduced game Continue doing so successively

46 Example: Two competing Bars Two bars (bar 1, bar 2) compete Can charge price of $2, $4, or $5 for a drink 6000 tourists pick a bar randomly 4000 natives select the lowest price bar $2$4$5 Bar 1 $2 10, 1014, 1214, 15 $4 12, 1420, 2028, 15 $5 15, 1415, 2825, 25 Bar 2 No dominant strategy for the both players.

47 Successive Elimination of Dominated Strategies $4$5 Bar 1 $4 20, 2028, 15 $5 15, 2825, 25 25, 28,15 14,15 $5$4 15,28 15,14 $5 20, 12,14 $4 Bar 1 14,1210, $2,,,,,,, Bar 1,, Bar 2

48 An example for Successive Elimination of strictly dominated strategies, or the process of iterated dominance

Equilibrium The interaction of all players' strategies results in an outcome that we call "equilibrium." Traditional applications of game theory attempt to find equilibria in games. In an equilibrium, each player is playing the strategy that is a "best response" to the strategies of the other players. No one is likely to change his strategy given the strategic choices of the others. Equilibrium is not: The best possible outcome. Equilibrium in the one-shot prisoners' dilemma is for both players to confess. A situation where players always choose the same action. Sometimes equilibrium will involve changing action choices (known as a mixed strategy equilibrium). 49

Definition: Nash Equilibrium “If there is a set of strategies with the property that no player can benefit by changing his/her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.” Source: 50

Nash equilibrium If each player has chosen a strategy and no player can benefit by changing his/her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. 51

52 No strictly dominant strategies and no strictly dominated strategies.

53 Finding Nash equilibria: (a) with strike-outs; (b) with underlinings

Prisoner’s Dilemma: finding Dominated Strategies 54 Which is a Nash Equilibrium?

Prisoner’s Dilemma : Applications Relevant to: – Nuclear arms races. – Dispute Resolution and the decision to hire a lawyer. – Corruption/political contributions between contractors and politicians. How do players escape this dilemma? – Play repeatedly – Find a way to ‘guarantee’ cooperation – Change payoff structure 55

Nuclear arms races prisoner's dilemma in disguise 56 Is there a Nash Equilibrium?

57 Cigarette Advertising prisoner's dilemma in disguise Philip Morris No AdAd Reynolds No Ad 50, 50 20, 60 Ad 60, 20 30, 30

Environmental policy prisoner's dilemma in disguise 58 Factory C pollutionNo pollution Factory R pollution 50, 50 60, 20 No pollution 20, 60 20, 20 Two factories producing same chemical can choose to pollute (lower production cost) or not to pollute (higher production cost).

Tragedy of the Commons Game theory can be used to explain overuse of shared resources. Extend the Prisoner’s Dilemma to more than two players. Each member of a group of neighboring farmers prefers to allow his cow to graze on the commons, rather than keeping it on his own inadequate land, but the commons will be rendered unsuitable for grazing if it is overgrazed. A cow costs a dollars and can be grazed on common land. The value of milk produced (f(c) ) depends on the number of cows on the common land. Per cow: f(c) / c 59

Tragedy of the Commons To maximize total wealth of the entire village: max f(c) – ac. – Adding another cow is exactly equal to the cost of the cow. What if each villager gets to decide whether to add a cow? Each villager will add a cow as long as the cost of adding that cow to that villager is outweighed by the gain in milk. 60

Tragedy of the Commons When a villager adds a cow: – Output per cow goes from f(c) /c to f(c+1) / (c+1) – Cost is a – His benefit is g(c)n where g(c)=f(c)/c – a is the gain per cow, and n is the number of cows he owns Each villager will add cows until output - cost = 0. Problem: each villager is making a local decision (will I gain by adding cows), but creating a net global effect (everyone suffers) 61

Tragedy of the Commons a form of payoff matrix 62 Your neighbor Add a cowDon’t add add a cow g(c+2)(n+1)g(c+1)(n+1) Don’t add g(c+1)n g(c)n As long as g is independent of c, adding cow is the dominant strategy for everybody.

63 Tragedy of the Commons more general form of payoff matrix Pay the cost Deny to pay B=benefit C<0 is the cost All others

Tragedy of the Commons Problem: cost of maintenance is externalized – Farmers don’t adequately pay for their impact. – Resources are overused due to inaccurate estimates of cost. Relevant to: – IT budgeting – Bandwidth usage, Shared communication channels – Health or other social benefits – Environmental laws, overfishing, whaling, pollution, etc. 64

Cost to press button = 2 units When button is pressed, food given = 10 units Another Example: Big & Little Pigs 65

Decisions, decisions What’s the best strategy for the little pig? Does he have a dominant strategy? Does the big pig have a dominant strategy?

67 Small Company researchNo research Big Company research 5, 1 4, 4 No research 9, -1 0, 0 Research in industries Big & Little Pigs in disguise

Maximin & Minimax Equilibrium in a zero-sum game Minimax - minimizing the maximum loss (loss-ceiling, defensive) Maximin - maximizing the minimum gain (gain-floor, offensive) Minimax = Maximin 68

Maximin, Minimax & Equilibrium Strategies 69

70 A zero-sum game with a saddle point. Saddle point Is this a Nash Equilibrium? MaxiMin MiniMax

The Minimax Theorem “Every finite, two-person, zero-sum game has a rational solution in the form of a pure or mixed strategy.” John Von Neumann, 1926 For every two-person, zero-sum game with finite strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2's strategy, the best payoff possible for player 1 is V, and (b) Given player 1's strategy, the best payoff possible for player 2 is −V. 71

Two-Person, Zero-Sum Games: Summary Represent outcomes as payoffs to row player Find any dominating equilibrium Evaluate row minima and column maxima If maximin=minimax, players adopt pure strategy corresponding to saddle point; choices are in stable equilibrium -- secrecy not required If maximin minimax, find optimal mixed strategy; secrecy essential 72

Summary: Look for any equilibrium Dominating Equilibrium Minimax Equilibrium Nash Equilibrium 73

Pure & mixed strategies 74 A pure strategy provides a complete definition of how a player will play a game. It determines the move a player will make for any situation they could face. A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. In a pure strategy a player chooses an action for sure, whereas in a mixed strategy, he chooses a probability distribution over the set of actions available to him.

All you need to know about Probability 75 If E is an outcome of action, then P(E) denotes the probability that E will occur, with the following properties: 1.0  P(E)  1 such that: If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 2.The probabilities of all the possible outcomes must sum to 1

Mixed strategies 76 Some games, such as Rock-Paper-Scissors, do not have a pure strategy equilibrium. In this game, if Player 1 chooses R, Player 2 should choose p, but if Player 2 chooses p, Player 1 should choose S. This continues with Player 2 choosing r in response to the choice S by Player 1, and so forth. In games like Rock-Paper-Scissors, a player will want to randomize over several actions, e.g. he/she can choose R, P & S in equal probabilities.

77 no No Nash equilibrium for pure strategy x y1-x-y x=probability to take action R y=probability to take action S 1-x-y=probability to take action P Mixed strategies

78 They have to be equal if expected payoff independent of action of player 2

Two-Person, Zero-Sum Game: Mixed Strategies Column Player: Row Player: Matrix of Payoffs to Row Player: 1212 A B Column Maxima: Row Minima: 79 No dominating strategy

Two-Person, Zero-Sum Game: Mixed Strategies Column Player: Row Player: Matrix of Payoffs to Row Player: 1212 A B Column Maxima: Row Minima: MaxiMin MiniMax No Saddle Point! MaxiMin MiniMax 80

Optimized Mixed Strategy: Graphical Solution 01 x VR 10 12/17 50/17 VR < 0*x+10(1-x) VR < 5x-2(1-x)= -2 +7x Optimal Solution: x=12/17, 1-x=5/17 VRMAX=50/17 2A 1A 1B 2B Probability of taking action 1 81

Graphical Solution 01 x VR 10 12/17 50/17 VR < 10(1-x) VR < -2 +7x y=1 y=0 y=.75 y=.5 y=.25 Optimal Solution: x=12/17, 1-x=5/17 VRMAX=50/ A 2B 1B 1A y= probability of taking action A

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x = probability taking action 1 1-x = probability taking action 2 84

85 Probability of player1 taking action 1 01 x 1B 1A 2A 2B Payoffs of player2 3/7 Optimal Solution: x=3/7, 1-x=4/7 VRMAX=4+4/7

86 x = probability taking action A 1-x = probability taking action B

87 Payoff of pure strategy Payoff of mixed strategy

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89 Pareto optimal Nash equilibrium

90 N-person games Larger games (More than 2 players) An Example of a 3-person non- cooperative game: Truel

A truel is like a duel, except that three players. Each player can either fire, or not fire, his or her gun at either of the other two players. The players’ preferences are: lone survival (the best = 4), survival with another player (the second best = 3), all players’ survival (the second worst=2), the players’ own death (worst case=1). If they have to make their choice simultaneously, what will they do? Ans. All of them will fire at either one of the other two players. If their choices are made sequentially (A>B>C>A>B>…) and the game will continue until only one player lift, what will they do? Ans. They will never shoot. 91

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Example: The paradox of the Chair’s Position Three voters ABC are electing the chairperson among them. Voter A has 3 votes. Voters B and C have 2 votes each. Voter A’s preference is (ABC). Voter B’s preference is (BCA). Voter C’s preference is (CAB). Who will win if voters vote their first preference? (sincere voting) Who will win if voters will consider what other players may do? (sophisticated voting) 93

If voters vote sincerely, Voters A will vote for voter A, voters B will vote for voter B, voters C will vote for voter C. So, the winner is voter A. Let’s consider voters A and BC as follows. A\ (BC)ABBBBC………. AABA BBBB CCBC So, the dominant strategy for voter A is voting for A. Assuming voter A will vote for A, let’s consider voters B and C. 94

B\CABC AAAA BABA CAAC So, the dominant strategy for voter C is voting for C. Assuming voters A and C will vote for A and C respectively, let’s consider voter B. B votes forABC resultAAC So, the dominant strategy for voter B is voting for C. As a result, voters A, B and C will vote for A, C and C, respectively. So, the winner is voter C. 95

Impact of game theory Nash earned the Nobel Prize for economics in 1994 for his “pioneering analysis of equilibria in the theory of non- cooperative games” Nash equilibrium allowed economist Harsanyi to explain “the way that market prices reflect the private information held by market participants” work for which Harsanyi also earned the Nobel Prize for economics in 1994 Psychologist Kahneman earned the Nobel prize for economics in 2002 for “his experiments showing ‘how human decisions may systematically depart from those predicted by standard economic theory’” 96

Fields affected by Game Theory Economics and business Philosophy and Ethics Political and military sciences Social science Computer science Biology 97