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## Game Theory

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**Game Theory**By Chattrakul Sombattheera Supervisors A/Prof Peter Hyland & Prof Aditya Ghose Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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) Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 = S1xS2x … xSn is the set of combinations of strategy profiles of n players • Utility function ui: S R of each player Coalition Formation Roadmap: Chattrakul Sombattheera**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} Coalition Formation Roadmap: Chattrakul Sombattheera**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 = Coalition Formation Roadmap: Chattrakul Sombattheera**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 = Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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? Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**Non-Zero Sum Game**• The sum of the utility is not zero • Prisoner Dilemma • Nash equilibrium • Chicken • Stag Hunt Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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*.” Coalition Formation Roadmap: Chattrakul Sombattheera**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) Coalition Formation Roadmap: Chattrakul Sombattheera**Nash Equilibrium**• What is the equilibrium in Prisoner Dilemma? • Usually, dominant equilibrium is Nash equilibrium • But, Nash Equilibrium may not be dominant equilibrium Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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) Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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? Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**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 !!! Coalition Formation Roadmap: Chattrakul Sombattheera**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. Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera**Equilibrium & Optimum**• In Stag Hunt, there are 2 equilibria (stag, stag) and (hare, hare) • Only one of the equilibria is optimal Coalition Formation Roadmap: Chattrakul Sombattheera**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 Coalition Formation Roadmap: Chattrakul Sombattheera