Game Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India February 2008

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1 Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India February 008 Chapter 3: Strategic Form Games Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send to hari@csa.iisc.ernet.in. A more thorough version would be available soon in this space. Strategies The notion of strategy is a key notion in game theory. A strategy is a complete contingent plan which specifies what a player will do at each of his information sets where he is called upon to play. Recall that the set of information sets of a player represents the set of all possible distinguishable circumstances in which the player may be required to play. A player s strategy completely specifies the action he is going to choose in each of his information sets, if it is reached during play of the game. Let IS i denote the collection of information sets of player i in a game. Let A be the set of possible actions in the game. Let J be an information set of player i, i.e., J IS i. Recall that C(J) = set of actions possible at information set J. A strategy of player i can be defined as a mapping s i : IS i A such that s i (J) C(J) J IS i. Note that a strategy of a player is a complete contingent plan that specifies an action for every information set of the player. The player can make a table with two columns, one for information sets and another for corresponding actions and a representative can take over and play the game using the table look-up. Example: Matching Pennies with Observation Information sets of player 1: IS 1 = {{v 1 }}. Information sets of player :IS = {{v }, {v 3 }} Strategies of Player 1: s 11 : s 1 : {v 1 } H {v 1 } T 1

2 player 1 player H v1 T player v v3 H T H T v4 v5 v6 v Figure 1: Game tree for matching pennies with observation Strategies of Player : s 1 : {v } H {v 3 } H s : {v } H {v 3 } T s 3 : {v } T {v 3 } H s 4 : {v } T {v 3 } T We can now represent the payoffs in the following way. 1 s 1 s s 3 s 4 s 11 1, +1 1, +1 +1, 1 +1, 1 s 1 +1, 1 1, +1 +1, 1 1, +1 N = {1, } S 1 = {s 11,s 1 } S = {s 1,s,s 3,s 4 } u 1 : S 1 S R u : S 1 S R defined as in the table above (N,S 1,S,u 1 ( ),u ( )) defines the strategic form or the normal form of the game. This is also abbreviated as (N,(S i ) i N,(u i ) i N ), written often as (N,(S i ),(u i )). This is a two player zero sum game. Example: Matching Pennies Without Observation IS 1 = {{v 1 }} IS = {{v,v 3 }}

3 player 1 H v1 T v player v3 H T H T v v v v Figure : Game tree for matching pennies without observation Player 1 has two strategies and player has two strategies. s 11 s 1 s 1 s : {v 1 } H : {v 1 } T : {v,v 3 } H : {v,v 3 } T The payoff matrix here is 1 s 1 s s 11 1, +1 +1, 1 s 1 +1, 1 1, +1 We now have a strategic form game (N,(S i ),(u i )) with N = {1, } S 1 = {s 11,s 1 } S = {s 1,s } u 1, u are as defined in the above table. This is again a two player zero sum game. Example: Matching Pennies with Simultaneous Moves This has the same normal form representation as that of the game without observation. Definition: Strategic Form Game A strategic form game G is a tuple G = (N,(S i ) i N,(u i ) i N ) where N = {1,,...,n} is a finite set of players; S 1,S,...,S n are the strategy sets of the players; and u i : S 1 S S n R for i = 1,,..., n are von Neumann - Morgenstern utility functions. 3

4 We often denote S = S 1 S S n. S is called the set of strategy profiles of the players. The strategies here are also called actions or pure strategies Note that the utility of a player depends not only on his own strategy but on an entire strategy profile. Utility functions are also called payoff functions. Every profile of strategies induces an outcome in the game. The idea behind the normal form representation is that a player s decision problem is to simply choose a strategy that he thinks will counter best the strategies adopted by the other players. Each player is faced with the problem and therefore the players can be thought of as simultaneously choosing their strategies from the respective sets S 1,S...,S n. One can view the play of a strategic game as follows: each player simultaneously writes down a chosen strategy on a piece of paper and hands it over to a referee who then computes the outcome and the utilities. Every extensive form game has a unique normal form representation. The uniqueness is up to any renaming or renumbering of strategies. We also immediately observe that a given normal form game may correspond to multiple extensive form games. For example, the extensive form game in Figure 3 has the same normal form representation as the matching pennies game with observation. player a b c d player 1 L R L R L R L R Figure 3: Another extensive form representation for matching pennies with observation 1 a b c d L 1, +1 1, +1 +1, 1 +1, 1 R +1, 1 1, +1 +1, 1 1, +1 4

5 An Important question: Does the normal form summarize all of the strategically relevant information? This question has been debated quite intensely by game theorists. In static games where all the players (statically) choose their actions at the same time (without observing the choices of other players), the normal form and the extensive form have the same representational power. This is not so in the case of dynamic games. Interpretations of Strategic Form Games Osborne and Rubinstein [1] provide the following two interpretations. Interpretation 1 A strategic game is a model of an event that occurs only once. Each player knows the details of the game and the fact that all players are rational The players choose their strategies simultaneously and independently Each player is unaware of the choices being made by the other players. Interpretation A player can form an expectation of the other players behavior on the basis of information about the way that the game or a similar game was played in the past. A strategic game models a sequence of plays of the game under the condition that there is no strategic link between the plays of the game. That is, a player who plays the game many times should only worry about his own instantaneous payoff and ignore the effects of his current action on the future behavior of the other players. A class of games called repeated games will have to be used if there is a strategic link between plays of a game. Examples of Normal Form Games Example 1: Matching Pennies with Simultaneous Moves Recall in this game that two players 1 and put down their respective rupee coins, heads up or tails up. If both the coins match (both heads or both tails), then player 1 pays 1 Rupee to player. Otherwise, player pays 1 Rupee to player 1. It is easy to see that: N = {1, } S 1 = S = {H,T } S = S 1 S = {(H,H),(H,T),(T,H),(T,T)} 5

6 The payoff matrix is given by 1 H T H 1, +1 +1, 1 T +1, 1 1, +1 This is a classic example of a two player zero-sum game. Example : BOS Game This is called the Battle of Sexes or the Batch or Stravinsky game. Proposed by Luce and Raiffa (1957). Two players 1 and wish to go out together for a music concert or a film. Player 1 prefers music concert and player prefers film. 1 M F M,1 0,0 F 0,0 1, This game captures a situation where the players want to coordinate but they have conflicting interests. The outcomes (M,F) and (F,M) can be ruled out. The question is: which one between (M,M) and (F,F) is the likely outcome? Example 3: A Coordination Game This game is similar to BOS but the two players now have preference for the same option, namely music concert. In this case, the natural outcome should be (M,M). Example 4: Hawk-Dove (Chicken) 1 M F M (,) (0,0) F (0,0) (1,1) There are two players who are fighting over a company/prey/property/etc. Each player can behave like a hawk or a dove. 1 Hawk Dove Hawk 3,3 4,1 Dove 1,4 0,0 The best outcome happens for a player who acts as a hawk while the other acts like a dove. 6

7 When both are hawks, the outcome is least desirable since nobody wins. When both are hawks, the outcome is better than if both were doves. What is a good prediction for this game? Example 5: Prisoner s Dilemma This is attributed to Raiffa (1951) and Flood and Dresher (195). This is one of the most extensively studied problems in game theory. Two individuals are arrested for allegedly committing a crime and are lodged in separate prisons. The district attorney (DA) interrogates them separately. The DA privately tells each prisoner that if he is the only one to confess, he will get a light sentence of 1 year in jail while the other would be sentenced to 10 years in jail. If each player confesses, then each one would get 5 years in jail. If neither confesses, then each would get years in jail. 1 NC C NC, 10, 1 C 1, 10 5, 5 How would the prisoners behave in such a situation? They would like to play a strategy that is best response to a (best) response strategy that the other player may adopt, the latter player also would like to play a best response to the other player s best response, and so on. Observation 1 C is each player s best strategy regardless of what the other player plays: u 1 (C,C) > u 1 (NC,C) u 1 (C,NC) > u 1 (NC,NC) u (C,C) > u (C,NC) u (NC,C) > u (NC,NC) Thus (C,C) is a natural prediction for this game. Observation Though (C,C) is a natural prediction, the outcome (NC, NC) is the best outcome jointly for the players. Prisoner s Dilemma is a classic example of a game where rational, intelligent behavior does not lead to a socially optimal result (Pareto efficient outcome). 7

8 Observation 3 Each prisoner has a negative effect on the other. When a prisoner moves away from (NC, NC) to reduce his jail term by 1 year, the jail term of the other player increases by 8 years. This is an example of what is called externality. Observation 4 NC is good for a player only if the other player also plays NC. Thus cooperation leads to a socially optimal outcome. Example 6: DA s Brother This is a modification of the PD problem in which prisoner 1 is a brother of the District Attorney. As a result, the DA is lenient towards prisoner 1. Example 7: Cold War 1 NC C NC 0, 10, 1 C 1, 10 5, 5 The countries have to decide whether they should emphasize spending on defence or on healthcare. Observation 1 Pakistan India Healthcare Defence Healthcare 10,10 10, 0 Defence 0, 10 0,0 Each player finds that defence is the best response whatever the other player plays. Observation Healthcare is good only if the other player plays healthcare. Observation 3 Predicted outcome for the game is (defence, defence) but (healthcare, healthcare) is socially optimal (Pareto efficient). Observation 4 Like the Prisoner s Dilemma problem, this is an example where rationality leads to an outcome that is not socially optimal. 8

9 Observation 5 If the players can cooperate, the outcome will be socially optimal. Example 9: ISP Routing Game - An Example of Prisoner s Dilemma This is based on an example discussed in the paper by Tardos and Vazirani []. Consider two ISPs (Internet Source Providers) A and B. There are two kinds of traffic we model in this example. The first type of traffic emanates from node a 1 belonging to ISP A and is targeted towards node b 1 belonging to ISP B. The second kind of traffic originates at node b belonging to ISP B and destined towards node a belonging to ISP A. See Figure. C and NC are two transit points or peering points or hand-off points or peering points or hand-off point S between the two ISPs. That is the traffic between the two ISP networks can only flow either through C or NC. Network of ISP A Network of ISP B a 1 C b Source Source a b 1 Destination for traffic from b NC Destination for traffic from a 1 Figure 4: A network with two ISPs The two source points of traffic, a 1 and b, are closer to the transit point C while the destination points a and b 1 are closer to the transit point NC. ISP A faces the problem of routing the traffic from a 1 and has two strategies. First strategy is to first route the traffic from a 1 to C and let ISP B takes care of the traffic from b to b 1. Call this strategy C. The second strategy is to transmit the traffic from a 1 internally to the transit point NC and then let ISP B route it from NC to b 1. Call this strategy NC. The strategy C minimizes the work for ISP A but this strategy makes ISP B work hard. On the other hand, strategy N C involves more work for ISP A and less work for ISP B. Similarly, ISP B also has two strategies C and NC for routing the traffic from b to a. 9

10 The C and NC are intuitively similar to the strategies C and NC in the prisoner s dilemma problem. Let the cost of routing traffic from a 1 to C, b to C, NC to a 1, and NC to a be 1 unit each. On the other hand, let the cost of routing traffic from a 1 to NC and b to NC be 3 units each. Then the following matrix provides a reasonable representation for the payoff matrix of this game. Example 9: Pollution Control Game A \ B NC C NC -4, -4-6, - C -, -6-5, -5 This is again based on an example discussed in the paper by Tardos and Vazirani []. First we discuss the case of two players and show its similarity ti the prisoner s dilemma problem. Next we extend it to multiple players. Consider two neighboring countries 1 and which are grappling with the problem of pollution. Each country has to decide whether or not to pass a legislation for controlling pollution. The two strategies are 0 and 1. Where 0 stands for undertaking pollution control measures and 1 indicates not undertaking pollution control measure. Assume that the cost of pollution control is 5 units while the cost of mitigating pollution effects is 1 unit. If a country decides to pay the strategy 0, then the cost incurred for this country is 5 units. On the other hand, if the country decides to pay 1, then it incurs a cost of 1 unit for mitigating the effects of pollution it causes. In addition, even the other country also incurs a cost of 1 unit for mitigating the effects of pollution caused by the first country. Under this situation, note that the strategy profile (0, 0) will mean that both countries will spend 5 units on pollution control and remain pollution free. The strategy profile (1, 1) will mean that both countries will pollute and hence incur a cost of units each (1 unit for pollution caused by the other country). 1 \ , -5-6, , -6 -, - The payoff matrix will be as shown. We can generalize this to the case of n countries as follows. N = {1,,...,n} S 1 = S =...,= S n = {0,1} u i (s i,s i ) = 5(1 s i ) j N s j Note that while u i (0,s i ) = 5 j i s j u i (1,s i ) = 1 j i s j 10

11 Thus u i (1,s i ) > u i (0,s i ) s i S i Example 10: Tragedy of the Commons The Tragedy of the Commons is a type of social paradox or social trap, often economic. The problem involves a conflict over resources between individual interests and social interests. A village has n farmers N = {1,,...,n}. Each farmer has the option of keeping a sheep or not. S 1 = S = = S n = {0,1}. Utility from keeping a sheep arises because of milk, wool, etc. Utility = 1. The village has grassland (limited) and when a sheep grazes on this, the damage to the environment = 5. The damage to the environment is to be shared equally by the farmers. Let s i be the strategy of each farmer. Then s i = 0,1. The payoff to farmer i is given by: For the case n =, the payoff matrix would be: Observation 1 = u i (s 1,s,...s i,...,s n ) [ ] 5(s1 + + s n ) = s i n ,0.5, ,.5 4, 4 If n > 5, keeping a sheep would add more utility to a farmer from milk/wool than subtract utility from him due to environmental damage. If n < 5, then the farmer gets less utility from keeping than from not keeping a sheep. If n = 5, the farmer has equal benefit and loss. Observation If n > 5, then every farmer would like to keep a sheep. How about n 5? Observation 3 If the Government now imposes a pollution tax of 5 units for every sheep kept, the payoff becomes: Now every farmer will prefer not to keep a sheep. u 1 (s 1,...,s n ) = s i 5s i 5(s s n ) n 11

12 Example 10: A Sealed Bid Auction The problem here is to allocate a unique object to one of n players in exchange for a payment. N = {1,,...,n} Sealed bids are sought from the n players. Let v 1,v,...,v n be the valuations of the players for the object. Then the strategy sets of the players are: S 1 = (0,v 1 ] S = (0,v ]. S n = (0,v n ] Assume that the object is awarded to the agent with the lowest index among those who bid the highest. Let b 1,b,...,b n be the bids from the n players. Then the allocation will be: = 1 if b i > b j j = 1,...,i 1 = 0 otherwise and b i b j j = i + 1,...,n Let p i (b 1,...,b n ) be the payment. Define the payoff to the players as u i (b 1,b,...,b n ) = x i (b 1,b,...,b n )(v i p i (b 1,...,b n )) The question is: what is the predicted outcome of this game. First Price Sealed Bid Auction Here the winner pays as he bids. Second Price Sealed Bid Auction Here, the payment by the winner is the highest bid among the players who do not win. Example 11: A Duopoly Pricing Game (Bertrand Model) This is due to Bertrand (1883). There are two companies 1 and which wish to maximize their profits. The demand for a price p is given by a continuous and strictly decreasing function x(p). The cost for producing each unit of product = c > 0. The companies simultaneously choose their prices p 1 and p. The amount of sales for each company is given by: x 1 (p 1,p ) = x(p 1 ) if p 1 < p = 1 x(p 1) if p 1 = p = 0 if p 1 > p x (p 1,p ) = x(p ) if p < p 1 = 1 x(p ) if p 1 = p = 0 if p > p 1 1

13 It is assumed that the firms incur production costs only for an output level equal to their actual sales. Given prices p 1 and p, the utilities of the two companies are: u 1 (p 1,p ) = (p 1 c)x 1 (p 1,p ) u (p 1,p ) = (p c)x (p 1,p ) Note that for this game, N = {1,} and S 1 = S = (0, ). Result: For the Bertrand duopoly model, the profile (c,c) gives the unique Nash equilibrium. The implication of this result is that in the equilibrium, the companies set their prices equal to the cost. Example 1: A Duopoly Pricing Game (Cournot Model) Here, the two companies simultaneously decide how much to produce. This model is due to Cournot (1838). An example of this is provided by agriculturists deciding how much of a perishable crop to pick each morning to send to a market. Here p( ) = x 1 ( ) is the inverse demand function and gives the price that clears the market. Here N = {1,} and S 1 = S = {1,,...,Q}, where Q is some maximum quantity. The utility functions are given by: Problems u 1 (q 1,q ) = p(q 1,q )q 1 cq 1 u (q 1,q ) = p(q 1,q )q cq. 1. (Mascolell, Whinston, Green [3]). The following payoff matrix corresponds to a modified version of the Prisoner s Dilemma problem called the DA s brother problem. In this problem prisoner 1 is related to the District Attorney. How is this problem different? Does it have a strongly dominant or a weakly dominant equilibrium? 1 NC C NC 0, 10, 1 C 1, 10 5, 5. (Attrition Game - Osborne and Rubinstein [1]) Two players are involved in a dispute over an object. The value of the object to player i is v i > 0. Time is modeled as a continuous variable that starts at 0 and runs indefinitely. Each player chooses when to concede the object to the other player; if the first player to concede does so at time t, the other player obtains the object at that time. If both players concede simultaneously, the object is split equally between them, player i receiving a payoff of v i /. Time is valuable: until the first concession each player loses one unit of payoff per unit time. Formulate this situation as a strategic game. 3. (A location game -Osborne and Rubinstein [1] ) Each of n people chooses whether or not to become a political candidate, and if so which position to take. There is a continuum of citizens, each of whom has a favorite position; the distribution of favorite positions is given by a density function f on [0,1] with f(x) > 0 for all x [0,1]. A candidate attracts the votes of those citizens whose favorite positions are closer to his position than to the position-of any other candidate; 13

14 if k candidates choose the same position then each receives the fraction 1/k of the votes that the the position attracts. The winner of the competition is the candidate who receives the most votes. Each person prefers to be the unique winning candidate than to tie for the first place, prefers to tie for the first place than to stay out of the competition, and prefers to stay out of the competition than to enter and lose. Formulate this situation as a strategic game. 4. (Guess the Average - Osborne and Rubinstein [1]) Each of n people announces a number in the set {1,...,K}. A prize of $1 is split equally between all the people whose number is closest to 3 of the average number. Formulate this as a strategic form game. 5. (An investment race - Osborne and Rubinstein [1]) Two investors are involved in a competition with a prize of $1. Each investor can spend any amount in the interval [0,1]. The winner is the investor who spends the most; in the event of a tie each investor receives $0.50. Formulate this situation as a strategic game. To Probe Further The material discussed in this chapter draws upon mainly from three sources, namely the books by Myerson [4], Mascolell, Whinston, and Green [3], and Osborne and Rubinstein [1]. The following books also contain illustrative examples of strategic form games: Osborne [1], Straffin [5], and Binmore [6]. References [1] Martin J. Osborne and Ariel Rubinstein. A Course in Game Theory. Oxford University Press, [] E. Tardos and V. Vazirani. Introduction to game theory. In Algorithmic Game Theory, pages Cambridge University Press, 007. [3] Andreu Mas-Collel, Michael D. Whinston, and Jerry R. Green. Micoreconomic Theory. Oxford University Press, [4] Roger B. Myerson. Game Theory: Analysis of Conflict. Harvard University Press, [5] Philip D. Straffin Jr. Game Theory and Strategy. The Mathematical Association of America, [6] Ken Binmore. Fun and Games : A Text On Game Theory. D. C. Heath & Company,