An introduction to Game Theory

An Introduction to Game Theory Presented as an undergraduate class inMultimedia Mathematics Paul Trafford paul.trafford@stx.oxon.org6 July 2011

Let’s Play a Game! Description: Bank has up to £1,000 to give away to the person or persons who choose the highest number. Players: Each individual student or group. Objective: To win as much as possible.  Rules. No communication between the players Choose a number N >= 1 and write it downon a piece of paper along with student/group name. The student(s) who chooses max. value of N wins total of £1,000/N 3

Let’s Play the Game Again! Description: Bank has up to £1,000 to give away to the person or persons who choose the highest number. Players: Each individual student or group. Objective: To win as much as possible.  Rules. Communication allowedbetween the players Choose a number N >= 1 and write it downon a piece of paper along with student/group name. The student(s) who chooses max. value of N wins total of £1,000/N 4

What is Game Theory? Definition of Game Theory The analysis of competitive situations (or situations of conflict) using mathematical models Essential Terminology The way a game is played depends on strategy– a plan of action before the game begins. A solutionis the adoption of a strategy that yields a particular outcome. Compare “solving” environmental problems with “solving” an equation. 5

What is it about? Fundamentally about the study of decision-making Investigations are concerned more with choices and strategies than ‘best’ solutions. It seeks to answer the questions: What strategies are there? What kinds of solutions are there? Examples: Chess, Go, economic markets, politics, elections, family relationships, etc. Characteristics of Game Theory 6

History (1) The study of games is many centuries old. More systematic developments in Game Theory took place in the first half of the 20th Century. Main FoundersJohn Von Neumann (mathematician) Oskar Morgenstern (economist) 7 Image sources: Los Alamos National Laboratory, http://www.lanl.gov/history/atomicbomb/images/NeumannL.GIFand American Mathematical Society, http://www.ams.org/samplings/feature-column/fcarc-rationality

History (2) Main publication: von Neumann & Morgenstern: Theory of Games and Economic Behaviour. Princeton University Press, 1944. Goal: Application of mathematical methods to broadly analyse games A new scientific approach to the study of economics. Applications: Aided by computers, theory has been broadly applied in large-scale operations such as international trade. 8

Game Theory is inter-disciplinary Game Theory 9

What makes a Game? Elements in a Game One or more players– participants, each may be an individual, a group or organisation, a machine, and so on. One or more moves(or choices) – where a move is an action carried out during the game, including chancemoves(when “nature plays a hand”) as in the toss of a coin. A set of outcomes– where an outcome is the result of the completion of one or more moves [e.g. game of chess may end in checkmate or a draw] Payoff– an amount received for a given outcome. Finally, a set of ruleswhich specify the conditions for the players, moves, outcomes and payoffs. 10

Strategy How should one play the game? Definition: A strategyis a plan of action by which a player has a decision rule to determine their set of moves for every possible situation in a game. A strategy is said to be pureif it at every stage in the game it specifies a particular move with complete certainty. A strategy is said to be mixed if it applies some randomisation to at least one of the moves. For each game, there are typically multiple pure strategies. Note that the randomisation is a set of fixed probabilities, where the sum of the probabilities is 1. 11

In this ‘game’ a ‘player’ is a commuter who is returning home from work – their objective is to return home as soon as possible. They can choose between train, bus and subway The first choice is ‘catch the train’, the second choice is ‘catch the bus’ and so on. • A commuter who always chooses to catch the train is following a ‘pure’ strategy. • A commuter who sometimes picks the train and sometimes the bus is following a ‘mixed’ strategy. Question: is this a one player game? Strategy: Travel Example 12 Photo credit: Nyao148 : Mejiro railway station http://en.wikipedia.org/wiki/File:Mejiro-Sta.JPG

Types of Games (1): Co-operative vs. Non co-operative Games Our first game (slide 2): non-cooperative Our second game (slide 3): cooperativeCooperation generally may lead to higher payoffs. Further Examples: Countries cooperate on trade (reduced tariffs) leading to boost in exports Two leading national social networking sites share technical knowledge and keep out an overseas competitor. Cartel: formation of monopoly by multiple organisations. 13

Types of Games (2): Perfect vs. Imperfection Information A game is said to have perfect Informationif all the moves of the game are known to the players when they make their move. Otherwise, the game has imperfect information. A large class of games of imperfect information are simultaneous games- games in which all players make their moves at the same time without knowing what the others will play. (The decisions may be made beforehand, but are not communicated). A game is said to be deterministic– if there are no chance moves. Otherwise, the game is non-deterministic. 14

A selection of games 15 Photo credits: Morten Johannes Ervik [Go], Jose Daniel Martinez [Chess], William Hartz(Scrabble), David ten Have (Ludo), WikiJET (Janken), Cyron Ray Macey(Tic Tac Toe), DaylandShannon (Monopoly), Denise Griffin (Bridge), Steve Snodgrass (Draughts)

How to classify? There are a number of [orthogonal] criteria that may be used as the basis for classifying games. A common one uses two: im/perfect information and chance/not chance. 16

Classification of games: 17 Photo credits: Morten Johannes Ervik [Go], Jose Daniel Martinez [Chess], William Hartz(Scrabble), David ten Have (Ludo), WikiJET (Janken), Cyron Ray Macey(Tic Tac Toe), DaylandShannon (Monopoly), Denise Griffin (Bridge), Steve Snodgrass (Draughts)

Zero vs. Non-Zero-Sum Games One of the most important classifications . A game is said to be zero-sumif wealth is neither created nor destroyed among the players. A game is said to be non-zero-sum if wealth may be created or destroyed among the players (i.e. the total wealth can increase or decrease). All examples above are zero-sum because they are competitive leisure games. However, most real-life situations are non-zero-sum (as indicated, for example, by how economies can grow). 18

PART B: Zero-Sum Games and Extended Form 19

1- Person Game: Tomato Plants (1) There are many 1 person games – including popular card games called ‘Patience’. They are instructive in decision-making. Example: Growing tomato plants…! 20 Photo credit: ManjithKainickarahttp://www.fotopedia.com/items/flickr-1061718736

1- Person Game: Tomato Plants (2) Objective: Grow a healthy tomato plant! Rules. One must make at least one move – plant a seed. Afterwards, one can make any number of moves: 21 Photo credit: ManjithKainickarahttp://www.fotopedia.com/items/flickr-1061718736

1- Person Game: Tomato Plants (3) How to Model? 22 Photo credit: ManjithKainickarahttp://www.fotopedia.com/items/flickr-1061718736

2- Person Game: Simple Nim (Also called the ‘subtraction game’) Rules Two players take turns removing objects from a single heap or pile of objects. On each turn, a player must remove exactly one or two objects. The winner is the one who takes the last object Demonstration: http://education.jlab.org/nim/index.html 23

Simplified Nim: winning strategy: proof Lemma: Suppose that Players A and B are playing the Nim subtraction game where at each move a player can remove between 1 and c counters, then a player has a winning strategy if they can play a move that leaves k(c+1) counters. Proof We prove this for Player A (1) Base Case (k=1): Suppose A leaves c+1 counters, then B has to choose to remove x:1≤x≤c. This implies that there are y = c+1-x left, where 1 ≤ y ≤ c. Then A chooses y and wins. 24

Simplified Nim: proof (2) (2) Inductive step: Assume the statement is true for k=n (n≥1). I.e. if Player A leaves n(c+1) , then player A wins. Suppose A leaves (n+1)(c+1) counters left, i.e. nc+n+c+1 If B chooses x:1≤x≤c, this leaves nc+n+c+1-x. Then A chooses c+1-x, leaving n(c+1). (3) Completion of proof by induction: Thus if the case k=n is true, then so is the case k=n+1 We have the base case k=1, is true, so the statement is true for k=2,3,… and so on. The Lemma is thus proved by induction for all values of k. 25

Simple Nim: Another go? Rules Two players take turns removing objects from a single heap or pile of objects. On each turn, a player must remove exactly one or two objects. The winner is the one who takes the last object Strategy Leave a multiple of 3. Demonstration: http://education.jlab.org/nim/index.html 26

2- Person Game: Traditional Nim (General form) Rules Two players take turns removing objects from distinct heaps or piles of objects. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap. Strategy: “To find out which move to make, let X be the Nim-sum of all the heap sizes. Take the Nim-sum of each of the heap sizes with X, and find a heap whose size decreases. The winning strategy is to play in such a heap, reducing that heap to the Nim-sum of its original size with X.” - Wikipedia entry 6/2011 27

Games in Extensive Form: Modelling by Trees We may model how the set of states in a game by using a tree with nodes and edges – called extensive form. Gambit is a set of software tools for doing computation on finite, non-cooperative games. It provides tree representations. Project founded in the mid-1980s by Richard McKelvey at the California Institute of Technology, USA. 28 [ Gambit Web site: http://www.gambit-project.org/ ]

Gambit Example: Tree for Nim (2,2) We may model how the set of states in a game by using a tree with nodes and edges. E.g. (2,2) game: 29

PART C: Zero-Sum Games in Normal Form 30

Introducing 2 person games in Normal Form We represent the players by Player A and Player B (or simply A and B) and denote the moves they can make as A1, A2, …, An and B1, B2, …, Bm respectively. These moves are made simultaneously, so these are games of imperfect information. We represent the game in normal form, i.e. using payoff matrices, where the value of each cell (i,j) is the payoff corresponding to the moves Ai and Bj respectively. 31

Normal Form: example of 2*2 game In the following example, we treat the special case where each player has 2 moves. (Note the payoffs are the values that will be given to Player A)Each row or column of payoffs is called an imputation. Player A has two moves: A1 and A2. Player B has two moves: B1 and B2. The payoff for a game is given by the intersection. Thus if the moves are respectively A1 and B2, then the payoff is zero. 32

Solutions of 2 person games A solutionis expressed as a set of strategies for all players that yields a particular payoff, generally the optimal payoff for both players. This payoff is called the valueof the game. Suppose, for example, each player adopts the strategy of choosing the move whose imputation contains the cell with the maximum payoff. Here, player A picks A2 as it contains a ‘4’, whereas player B selects B2 as it contains -2. This would yield 2 for player B. However, this is not a solution as it is not optimal for player A – they could always do better by playing A1. So the value of the game is >-2. 33

The Concept of Equilibrium (Pure Strategies) 1/2 So what strategies may yield optimal payoffs for both? Key concept: In an equilibrium, each player of the game has adopted a strategy that cannot improve his outcome, given the others' strategy. The method for this is: Player A considers each imputation and what is the least payoff value that may be gained by choosing that imputation. Similarly, Player B considers each imputation and what is the greatest payoff value that may be gained by choosing that imputation. 34

The Concept of Equilibrium (Pure Strategies) 2/2 Formally, this is the maximin criteria given by 𝑣𝐿=𝑚𝑎𝑥𝑖 𝑚𝑖𝑛𝑗 𝑒𝑖𝑗 𝑣𝑀=𝑚𝑖𝑛𝑗 𝑚𝑎𝑥𝑖 𝑒𝑖𝑗 (where 𝑒𝑖𝑗 denotes payoff in cell (i,j)). Example So, player A can expect to gain at least vL =2 Player B can expect to lose at most vM =3. 35

The Concept of Equilibrium (Pure Strategies): Saddle Points In the case that the value of the game is vL= vM , a saddle pointis any cell whose payoff is this value. Example Playing A1 => payoff of at least 0 Playing B2 => payoff of no more than 0 There is a unique saddle point – cell (A1,B2). If either player deviates from this, then they will do worse. Here, vL= vM = 0. 36

When there is no Saddle Point Consider again the following payoff matrix: We have seen above that the value of the game lies between 2 and 3. But, if player A always plays A2, then B can always play B2 and the payoff is 2, whereas is player A always plays A1, then B1 can always play B1, yielding 1, which is less than 2! Can player A gain more than 2…? Yes, because the game is of imperfect information – players don’t know each other’s move, but this means that we should not be predictable. 37

Simplification using Dominance For larger matrices, we may often simplify. The main technique for simplification is to compare pairs of columns, C and C’, say, and delete those columns where the payoff in C is always greater than that in C’ or vice versa. In this case we say C dominatesC’. (Similarly for rows).Thus, B4 dominates B1, B3 and B5, yielding :This matrix yields a saddle point corresponding to the moves A2 and B4, with value of the game=3. 38

Simplification using Dominance: Demonstration in Gambit 39

Mixed Strategies: Expectation 1/2 Scenario: Game is played repeatedly. In this case choosing the same pure strategy is not always optimal, so we can vary these pure strategies. To determine how we vary the strategies, we can apply probability theory. Key concept is Expectation:= the product of the probability of the occurrence of an event and the value associated with the occurrence of a given event. A player can use a mixed strategy– this is more than one pure strategy, where each pure strategy is played randomly according to a fixed probability yielding an expected payoff. 40

Mixed Strategies: Expectation 1/2 We then can determine the expected value of a game. Formally, as before, we denote the moves available to Players A and B as A1,A2, … ,An and B1,B2, … ,Bm respectively. Suppose the moves in A’s mixed strategy are played with probabilities x= (x1, x2, …, xn); and for B, y = (y1, y2, …, ym). Suppose the payoffs are given by 𝑒𝑖𝑗. Then the game’s expected value for A is E(x), where 𝐸𝒙=𝑖,𝑗𝑥𝑖𝑦𝑗𝑒𝑖𝑗 The expectation is the same for Player B. 41

Mixed Strategies: Expectation: Examples The moves available to Player A are A1,A2 and to Player B: B1,B2. Suppose the moves in A’s mixed strategy are played with probabilities x= (x1, x2); and for B, y = (y1, y2). Suppose the payoffs are given by 𝑒𝑖𝑗. Then the game’s expected value, E(x)=x1y1*1 + x1y2*4 + x2y1 *3 + x2y2*2 So what should the values be for x andy … ? 42

Minimax: The Concept of Equilibrium for Mixed Strategies Minimax– one of the key theories developed by Von Neumann and Morgenstern originally defined this only for zero-sum games Rationale: whatever the other player does, this return is assured on average. Theorem. In a two-person zero-sum game where player A has n strategies and player B has m strategies (where n and m are finite), then the minimax value of the game, v, is given by: v = max𝑥∈𝑋min𝑦∈𝑌𝑒(𝐱,𝒚)= max𝑦∈𝑌min𝑥∈𝑋𝑒(𝒙,𝒚) (The saddle point is a special case where xi = 1 for some i, yj=1 for some j.) Thus the solution is to play moves in fixed proportion x and the value can be determined by simply considering the expectation against any single move. 43

Minimax: Determination of the Mixed Strategies Determination of x and y Determine if there are any saddle points. If found then we have the solution and can stop here. Remove all dominated imputations (rows/columns), leaving a payoff matrix M. For the two players, solve Mx = v and MTy = v respectively, where v is a vector where each entry is v, the value of the game. (MT is the transpose of M) 44

Minimax Example (1/4) 𝑀=1432, 𝑀𝑇=1342 There are no saddle points, and no cases of dominance. Player B: 1432𝑥′𝑦′=𝑣𝑣Player A: 1342𝑥𝑦=𝑣𝑣 45

Minimax Example (2/4): Player A’s mixed strategy Let x:= the probability Player A plays A1Let y:= the probability Player A plays A1Then x+y=1. 1*x+ 3*y = v 4*x+ 2*y = v Therefore, from (1), x=v-3y. Substitute in (2) to give: 4(v-3y)+2y=v. Therefore, 3v=10y. Hence, 3x = 3(v-3y) = 10y-9y = y.Therefore x=0.25,y=0.75 and v=2.5 46

Minimax: Example (3/4): Player B’s mixed strategy Hence, for player B: Let x’:= the probability Player B plays B1Let y’:= the probability Player B plays B1Then x’+y’ = 1. 1*x’ + 4* y’ = 2.5 3*x’ + 2* y’ = 2.5 Therefore, 3(2.5-4y’)+2y’= 2.5 Hence, 5 = 10y’ => y’ = 0.5. Therefore x’ = 0.5 47

Minimax Example (4/4) – use of Gambit Gambit provides modelling of games in normal form – Gambit calls them “strategic games”. In the screenshot, each cell has a pair of payoffs - the first is what Player A receives, the second is what Player B receives. (Gambit is designed for non-zero-sum games – see later sections). It can compute the expected value and the corresponding equilibria mixed strategies of the two players. 48

Minimax Limitations Whilst the Minimax theorem provides a solution, it’s macro-oriented, i.e. not sensitive to individual variations. Thus ,[object Object]

Assumes repeated play and is a result that is more reliable the more times played

In practice, it takes no account of the strategy of the opponent – even if they keep playing the same pure strategy, the expected return is no more, no less…

The optimisation reflects a collective philosophy that markets find their natural level.49

An Overview of Non-Zero-Sum Games [Recap] A game is said to be non-zero-sum if wealth may be created or destroyed among the players (i.e. the total wealth can increase or decrease). In general, unlike for zero-sum games, in non-zero-sum games, wealth can be mutually created through cooperation. Cooperation may be achieved whether or not there is direct communication. Where there is no communication, information is necessarily imperfect. Where there is communication, there may be bargaining. 51

Analysis of Non-Zero-Sum Games Methods of mathematical logical, such as use of induction, are effective for determining strategies in Zero-sum games with perfect information. However they are less so for games of imperfect information, and are often not applicable to non-zero games. IF some assumptions are made THEN some mathematical techniques may be effectively applied. Prerequisites: Understand the environment, understand the individual and collective psychology (Thus we are moving from the domain of pure mathematics to embrace social sciences, particularly psychology and economics.) 52

Utility Payoffs are given as utility – the perceived worth of something Utility is a key concept and is determined by social and psychological factors. They depend upon personal preferences The same material payoff may have different utility (In economics, personal preference is often reckoned in terms of ranking a selection of consumer offerings. [Economic] agents are said to be “rational” if this ranking system is complete.) 53

Utility – Example (Exercise) Which would you choose? (Game is only played once!) 10 million Yen 100% chance 100 million Yen 20% chance 54

Utility – Example (Analysis) Expected return option (1) = 10 million yen, Expected return option (2) = 20 million yen, But option (1) has already great utility – utility curve may be logarithmic Here, if you have many friends playing or many attempts, then you should go for option 2. This is similar to philosophy of ‘penny shares’ – small investment, unlikely to succeed, but if it succeeds then it could be very successful. 55

Analytical Approaches to Non-Zero-Sum Games As before, the mathematical approaches use linear algebra, matrices, and probability theory. Hence the basic Concepts in Non-Zero-Sum Games: One-off vs Repeated games Payoff matrix Expectation Strategies – pure and mixed However, the generation of appropriate models requires Social Science tools that take account of the psychology of human behaviour, individual and collective; the analysis of markets, negotiation and bargaining. 56

Introducing The Prisoner’s Dilemma Description: Two men suspected of committing a bank robbery together and are arrested by the police. They are placed in separate cells, so cannot communicate. Each suspect may either confess or remain silent. They know the consequences of their actions. Suppose we call them Player A and Player B: ,[object Object]

If both A and B confess, then they go to prison for 5 years.If both A and B remain silent, then they go to prison for 1 year for carrying concealed weapons. 57

The Prisoner’s Dilemma: Payoff Matrix Non-zero-sum games of normal form may be represented by a payoff matrix, where each cell is an n-tuple, a set of payoffs, 1 for each player. Thus for the Prisoner’s Dilemma, a 2-person game, we have pairs of payoffs. If A1 denotes ‘Player A remains silent’, A2 denotes ‘Player A confesses’ (similarly for B), then we can represent the problem by the following matrix: 58

The Prisoner’s Dilemma: Strategy Player A reasons as follows: If Player B chooses B1, then I am better off choosing A2 (because O > -1). If Player B chooses B2, then I am better off choosing A2 (because -5 > -10). Similarly, for player B. Hence A2,B2 are selected. In fact, this reflects accepted theory: John Nash extended the minimax result of zero-sum games to non-zero-sum games. Informally, it states that a pair of mixed strategies is in Nash equilibrium, if it means that any unilateral (one-sided) deviation for either player would yield a payoff that was no more than the value of the pair. (A2,B2) are in equilibrium. Payoffs are (-5.-5) 59

Mixed Strategies for Non-zero-sum Games: Nash Equilibrium As mentioned above, John Nash’s theorem states that a pair of mixed strategies is in equilibrium if any unilateral (one-sided) deviation for either player would yield a payoff that was no more than the value of the pair. Formally, Definition. A pair of strategies, x*∈X, y*∈Y is an equilibrium pair for a non-zero-sum game if for any x∈X and y∈Y, eA(x,y*) ≤ e(x*,y*) and eB(x*,y) ≤ e(x*,y*) , where eAis player A’s payoff and eB is player B’s payoff. Theorem. Any two-person (zero-sum or non-zero-sum) with a finite number of pure strategies has at least one equilibrium pair. (Such a pair is called a Nash Equilibrium pair. Determining the solution is not trivial.) 60

The Prisoner’s Dilemma: Paradox Paradox: both players confess and spend 5 years in prison, whereas if they had remained silent they would have spent 1 year each in prison! Diagnosis: the unilateral view is not optimal. A bilateral (two-sided) view – involving cooperation – would suggest the other move for both players. This is covered by the notion of strategies being pareto optimal– when there is no other strategy in which both players are at least as well off. 61

The Repeated Prisoner’s Dilemma: Web demos There are many online versions of the Prisoner’s Dilemma. See e.g. Lessons from the Prisoner’s Dilemma: An interactive tutorial by Martin Poulter, April 2003, Economics Networkhttp://www.economicsnetwork.ac.uk/archive/poulter/pd.htm 62

The Prisoner’s Dilemma: Applications (1) What is it useful for? Usefulness usually determined by consideration of repeated games… Lessons for military (consider safety of the citizens of two rival powers: which is safer? If they both disarm (cooperative strategy)? Or if they are both heavily armed? Marketing strategies – if two rival companies both offer small discounts then they may receive many customers and retain a good market share. What if they offer huge discounts? 63

The Prisoner’s Dilemma: Applications (2) “In economics as in other realms of the prisoner's dilemma, success requires a willingness not to measure oneself against any one opponent. ''You do tend to compare yourself to other people,'' Dr. Hauser said. ''However, it turns out that if I do that I'm hurting myself very badly.'' Biological Applications” “In real life, that is, does cooperation depend on an internal sense of morality? Or does it depend on the complicated dynamics of environments where people challenge each other, betray each other and trust each other over and over again?” NY Times, PRISONER'S DILEMMA HAS UNEXPECTED APPLICATIONS By JAMES GLEICK Published: June 17, 1986 64

The Battle of the Sexes Suppose that a newlywed couple are both planning an outing at the weekend. They haven’t yet decided what to do. The husband would like to watch football, whereas the wife would like to go to a concert, but they would both prefer to be in the company of their spouses rather than go their separate ways. Suppose option 1 is football and option 2 is concert. Then the payoff matrix may look like this: 65

The Battle of the Sexes: Equilibria (Gambit) Gambit can calculate the equilibria and gives 3 of them: Two of the three are indicating cooperation 66

The Battle of the Sexes: Modelling in Maxima Maxima can be used to plot regions. Suppose the husband chooses to play option H1 with probability x. Therefore they play option H2 with probability 1-x. Similarly, the wife plays option W1 with probability y and option W2 with probability 1-y. We then can define expectation for each player as functions E1 and E2 respectively in variables x and y: ,[object Object]

E2:=(1*xy + 0*x(1-y)) + (0*(1-x)y + 4*(1-x)(1-y))Hence ,[object Object],67

An introduction to Game Theory

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (11)

Similar to An introduction to Game Theory

Similar to An introduction to Game Theory (20)

Recently uploaded

Recently uploaded (20)

An introduction to Game Theory

Editor's Notes