# 6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games

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1 6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4,

2 Introduction Outline Decisions, utility maximization Strategic form games Best responses and dominant strategies Dominated strategies and iterative elimination of strictly dominated strategies Nash Equilibrium Examples Reading: Fudenberg and Tirole, Chapter 1. 2

3 Introduction Motivation In many social and engineered systems, agents make a variety of choices. For example: How to map your drive in the morning (or equivalently how to route your network traffic). How to invest in new technologies. Which products to buy. How to evaluate information obtained from friends, neighbors, coworkers and media. In all of these cases, interactions with other agents affect your payoff, well-being, utility. How to make decisions in such situations? multiagent decision theory or game theory. 3

4 Introduction Rational Decision-Making Powerful working hypothesis in economics: individuals act rationally in the sense of choosing the option that gives them higher payoff. Payoff here need not be monetary payoff. Social and psychological factors influence payoffs and decisions. Nevertheless, the rational decision-making paradigm is useful because it provides us with a (testable) theory of economic and social decisions. We often need only ordinal information; i.e., two options a and b, and we imagine a preference relation that represents the ranking of different options, and we simply check whether a b or a b. But in game theory we often need cardinal information because decisions are made under natural or strategic uncertainty. The theory of decision-making under uncertainty was originally developed by John von Neumann and Oskar Morgenstern. 4

5 Introduction Decision-Making under Uncertainty von Neumann and Morgenstern posited a number of reasonable axioms that rational decision-making under uncertainty should satisfy. From these, they derived the expected utility theory. Under uncertainty, every choice induces a lottery, that is, a probability distribution over different outcomes. E.g., one choice would be whether to accept a gamble which pays \$10 with probability 1/2 and makes you lose \$10 with probability 1/2. von Neumann and Morgenstern s expected utility theory shows that (under their axioms) there exists a utility function (also referred to as Bernoulli utility function) u (c), which gives the utility of consequence (outcome) c. Then imagine that choice a induces a probability distribution F a (c) over consequences. 5

6 Introduction Decision-Making under Uncertainty (continued) Then the utility of this choice is given by the expected utility according to the probability distribution F a (c): U (a) = u (c) df a (c). In other words, this is the expectation of the utility u (c), evaluated according to the probability distribution F a (c). More simply, if F a (c) is a continuous distribution with density f a (c), then U (a) = u (c) f a (c) dc, or if it is a discrete distribution where outcome outcome c i has probability p i a (naturally with i p i a = 1), then a U (a) = p i u (c i ). i 6

7 Introduction Decision-Making under Uncertainty (continued) Given expected utility theory and our postulate of rationality, single person decision problems are (at least conceptually) simple. If there are two actions, a and b, inducing probability distributions F a (c) and F b (c), then the individual chooses a over b only if U (a) = u (c) df a (c) U (b) = u (c) df b (c). 7

8 Introduction From Single Person to Multiperson Decision Problems But in a multi-agent situation, the utility of an agent or probability distribution over outcomes depends on actions of others. A simple game of partnership represented as a matrix game: Player 1 \ Player 2 work hard shirk work hard (2, 2) ( 1, 1) shirk (1, 1) (0, 0) Here the first number is the payoff to player (partner) 1 and the second number is the payoff to player 2. More formally, the cell indexed by row x and column y contains a pair, (a, b) where a = u 1 (x, y ) and b = u 2 (x, y ). These numbers could be just monetary payoffs, or it could be inclusive of social preferences (the fact that you may be altruistic towards your partner or angry at him or her). Should you play work hard or shirk? 8

9 Strategic Form Games Strategic Form Games Let us start with games in which all of the participants act simultaneously and without knowledge of other players actions. Such games are referred to as strategic form games or as normal form games or matrix games. For each game, we have to define The set of players. The strategies. The payoffs. More generally, we also have to define the game form, which captures the order of play (e.g., in chess) and information sets (e.g., in asymmetric information or incomplete information situations). But in strategic form games, play is simultaneous, so no need for this additional information. 9

10 Game Theory: Lecture 2 Strategic Form Games (continued) Strategic Form Games More formally: Definition (Strategic Form Game) A strategic forms game is a triplet I, (S i ) i I, (u i ) i I such that I is a finite set of players, i.e., I = {1,..., I }; S i is the set of available actions for player i; s i S i is an action for player i; u i : S R is the payoff (utility) function of player i where S = i S i is the set of all action profiles. s S (s i i i, In addition, we use the notation = [s j ] j=i : vector of actions for all players except i. = j=i S j is the set of all action profiles for all players except i s i ) S is a strategy profile, or outcome. 10

11 Strategic Form Games Strategies In game theory, a strategy is a complete description of how to play the game. It requires full contingent planning. If instead of playing the game yourself, you had to delegate the play to a computer with no initiative, then you would have to spell out a full description of how the game would be played in every contingency. For example, in chess, this would be an impossible task (though in some simpler games, it can be done). Thinking in terms of strategies is important. But in strategic form games, there is no difference between an action and a pure strategy, and we will use them interchangeably. 11

12 Strategic Form Games Finite Strategy Spaces When the S i is finite for all i, we call the game a finite game. For 2 players and small number of actions, a game can be represented in matrix form. Recall that the cell indexed by row x and column y contains a pair, (a, b) where a = u 1 (x, y ) and b = u 2 (x, y ). Example: Matching Pennies. Player 1 \ Player 2 heads tails heads ( 1, 1) (1, 1) tails (1, 1) ( 1, 1) This game represents pure conflict in the sense that one player s utility is the negative of the utility of the other player. Thus zero sum game. More generally true for strictly competitive games, that is, games in which whenever one player wins the other one loses, though the sum of the payoffs need not be equal to 0. 12

13 Strategic Form Games Infinite Strategy Spaces Example: Cournot competition. Two firms producing a homogeneous good for the same market. The action of a player i is a quantity, s i [0, ] (amount of good he produces). The utility for each player is its total revenue minus its total cost, u i (s 1, s 2 ) = s i p(s 1 + s 2 ) cs i where p(q) is the price of the good (as a function of the total amount q), and c is unit cost (same for both firms). Assume for simplicity that c = 1 and p(q) = max{0, 2 q} Consider the best response correspondence for each of the firms, i.e., for each i, the mapping B i (s i ) : S i S i such that B i (s i ) arg max u i (s i, s i ). s i S i Why is this a correspondence not a function? When will it be a function? 13

14 Cournot Competition (continued) Strategic Form Games By using the first order optimality conditions, we have B i (s i ) = arg max (s i (2 s i s i ) s i ) s i 0 1 s i if s i 1, = 2 0 otherwise. s 2 1 1/2 B 1 (s 2 ) 1/2 s 1 B 2 (s 1 ) 1 The figure illustrates the best response correspondences (which in this case are functions). Assuming that players are rational and fully knowledgeable about the structure of the game and each other s rationality, what should the outcome of the game be? 14

15 Dominant Strategies Dominant Strategies Example: Prisoner s Dilemma. Two people arrested for a crime, placed in separate rooms, and the authorities are trying to extract a confession. prisoner 1 / prisoner 2 Confess Don t confess Confess ( 4, 4) ( 1, 5) Don t confess ( 5, 1) ( 2, 2) What will the outcome of this game be? Regardless of what the other player does, playing Confess is better for each player. The action Confess strictly dominates the action Don t confess Prisoner s dilemma paradigmatic example of a self-interested, rational behavior not leading to jointly (socially) optimal result. 15

16 Dominant Strategies Prisoner s Dilemma and ISP Routing Game Consider two Internet service providers that need to send traffic to each other Assume that the unit cost along a link (edge) is 1 s1 ISP1: s1 ISP2: s2 t1 t2 DC C Peering points t2 t1 s2 This situation can be modeled by the Prisoner s Dilemma payoff matrix. ISP 1 / ISP 2 Hot potato Cooperate Hot potato ( 4, 4) ( 1, 5) Cooperate ( 5, 1) ( 2, 2) 16

17 Dominant Strategies Dominant Strategy Equilibrium Compelling notion of equilibrium in games would be dominant strategy equilibrium, where each player plays a dominant strategy. Definition (Dominant Strategy) A strategy s i S i is dominant for player i if u i (s i, s i ) u i (s i, s i ) for all s i S i and for all s i S i. Definition (Dominant Strategy Equilibrium) A strategy profile s is the dominant strategy equilibrium if for each player i, s i is a dominant strategy. These notions could be defined for strictly dominant strategies as well. 17

18 Dominant Strategies Dominant and Dominated Strategies Though compelling, dominant strategy equilibria do not always exist, for example, as illustrated by the partnership or the matching pennies games we have seen above. Nevertheless, in the prisoner s dilemma game, confess, confess is a dominant strategy equilibrium. We can also introduce the converse of the notion of dominant strategy, which will be useful next. Definition (Strictly Dominated Strategy) A strategy s i S i is strictly dominated for player i if there exists some s i S i such that u i (s i, s i ) > u i (s i, s i ) for all s i S i. 18

19 Dominant Strategies Dominated Strategies Definition (Weakly Dominated Strategy) A strategy s i S i is weakly dominated for player i if there exists some s i S i such that u i (s i, s i ) u i (s i, s i ) for all s i S i, u i (s i, s i ) > u i (s i, s i ) for some s i S i. No player should play a strictly dominated strategy Common knowledge of payoffs and rationality results in iterated elimination of strictly dominated strategies 19

20 Dominant Strategies Iterated Elimination of Strictly Dominated Strategies Example: Iterated Elimination of Strictly Dominated Strategies. prisoner 1 / prisoner 2 Confess Don t confess Suicide Confess ( 2, 2) (0, 3) ( 2, 10) Don t confess ( 3, 0) ( 1, 1) (0, 10) Suicide ( 10, 2) ( 10, 0) ( 10, 10) No dominant strategy equilibrium; because of the additional suicide strategy, which is a strictly dominated strategy for both players. No rational player would choose suicide. Thus if prisoner 1 is certain that prisoner 2 is rational, then he can eliminate the latter s suicide strategy, and likewise for prisoner 2. Thus after one round of elimination of strictly dominated strategies, we are back to the prisoner s dilemma game, which has a dominant strategy equilibrium. Thus iterated elimination of strictly dominated strategies leads to a unique outcome, confess, confess thus the game is dominance solvable. 20

21 Dominant Strategies Iterated Elimination of Strictly Dominated Strategies (continued) More formally, we can follow the following iterative procedure: Step 0: Define, for each i, S0 i = S i. Step 1: Define, for each i, 1 0 S i = s i S i s 0 i S i s.t. u i s 0 i, s i > u i (s i, s i ) s i S i.... Step k: Define, for each i, S k k 1 = s i S i s k 1 S i s.t. u i s k 1 i i i, s i > u i (s i, s i ) s i S i. Step : Define, for each i, k S i = k=0 S i. 21

22 Dominant Strategies Iterated Elimination of Strictly Dominated Strategies (continued) Theorem Suppose that either (1) each S i is finite, or (2) each u i (s i, s i ) is continuous and each S i is compact (i.e., closed and bounded). Then S i (for each i) is nonempty. Proof for part (1) is trivial. Proof for part (2) in homework. Remarks: Note that S i need not be a singleton. Order in which strategies eliminated does not affect the set of strategies that survive iterated elimination of strictly dominated strategies (or iterated strict dominance): also in the homework. 22

23 Dominant Strategies How Reasonable is Dominance Solvability At some level, it seems very compelling. But consider the k- beauty game. Each of you will pick an integer between 0 and 100. The person who was closest to k times the average of the group will win a prize. How will you play this game? And why? 23

24 Revisiting Cournot Competition Dominant Strategies Apply iterated strict dominance to Cournot model to predict the outcome s 2 1 s 2 1 B 1 (s 2 ) B 1 (s 2 ) 1/2 1 4 / B 2 (s 1 ) 1/2 1 4 / B 2 (s 1 ) 1 4 / 1/ / 1/2 1 s 1 s 1 One round of elimination yields S 1 1 = [0, 1/2], S 2 1 = [0, 1/2] Second round of elimination yields S 1 2 = [1/4, 1/2], S 2 2 = [1/4, 1/2] It can be shown that the endpoints of the intervals converge to the intersection Most games not solvable by iterated strict dominance, need a stronger equilibrium notion 24

25 Nash Equilibrium Pure Strategy Nash Equilibrium Definition (Nash equilibrium) A (pure strategy) Nash Equilibrium of a strategic game I, (S i ) i I, (u i ) i I is a strategy profile s S such that for all i I u i (s i, s i ) u i (s i, s i ) for all s i S i. Why is this a reasonable notion? No player can profitably deviate given the strategies of the other players. Thus in Nash equilibrium, best response correspondences intersect. Put differently, the conjectures of the players are consistent: each player i chooses s i expecting all other players to choose s i, and each player s conjecture is verified in a Nash equilibrium. 25

26 Nash Equilibrium Reasoning about Nash Equilibrium This has a steady state type flavor. In fact, two ways of justifying Nash equilibrium rely on this flavor: 1 2 Introspection: what I do must be consistent with what you will do given your beliefs about me, which should be consistent with my beliefs about you,... Steady state of a learning or evolutionary process. An alternative justification: Nash equilibrium is self-reinforcing If player 1 is told about player 2 s strategy, in a Nash equilibrium she would have no incentive to change her strategy. 26

27 Nash Equilibrium Role of Conjectures To illustrate the role of conjectures, let us revisit matching pennies Player 1 \ Player 2 heads tails heads ( 1, 1) (1, 1) tails (1, 1) ( 1, 1) Here, player 1 can play heads expecting player 2 to play tails. Player 2 can play tails expecting player 1 to play tails. But these conjectures are not consistent with each other. 27

28 Intersection of Best Responses Nash Equilibrium Recall the best-response correspondence B i (s i ) of player i, B i (s i ) arg max u i (s i, s i ). s i S i Equivalent characterization: an action profile s is a Nash equilibrium iff s i B i (s i ) for all i I. Therefore, in Cournot as formulated above, unique Nash equilibrium. s 2 1 1/2 B 1 (s 2 ) B 2 (s 1 ) 1/2 s 1 Remark: When iterated strict dominance yields a unique strategy profile, is this necessarily a Nash equilibrium? unique Nash equilibrium? 1 28

29 Examples Example: The Partnership Game Let us return to the partnership game we started with. Player 1 \ Player 2 work hard shirk work hard (2, 2) ( 1, 1) shirk (1, 1) (0, 0) There are no dominant or dominated strategies. Work hard is a best response to work hard and shirk is a best response shirk for each player. Therefore, there are two pure strategy Nash equilibria (work hard, work hard) and (shirk, shirk). Depending on your conjectures ( expectations ) about your partner, you can end up in a good or bad outcome. 29

30 Examples Focal Points What do we do when there are multiple Nash equilibria? Our models would not be making a unique prediction. Two different lines of attack: Think of set valued predictions i.e., certain outcomes are possible, and Nash equilibrium rules out a lot of other outcomes. Think of equilibrium selection. Equilibrium selection is hard. Most important idea, Schelling s focal point. Some equilibria are more natural and will be expected. Schelling s example: ask the people to meet in New York, without specifying the place. Most people will go to Grand Central. Meeting at Grand Central, as opposed to meeting at any one of thousands of similar places, is a focal point. 30

31 Examples Examples: Battle of the Sexes and Matching Pennies Example: Battle of the Sexes (players wish to coordinate but have conflicting interests) Player 1 \ Player 2 ballet football ballet (1, 4) (0, 0) football (0, 0) (4, 1) Two Nash equilibria, (Ballet, Ballet) and (Soccer, Soccer). Example: Matching Pennies. Player 1 \ Player 2 heads tails heads ( 1, 1) (1, 1) tails (1, 1) ( 1, 1) No pure Nash equilibrium (but we will see in the next lecture that there exists a unique mixed strategy equilibrium). 31

32 Examples: Cournot Competition Examples We now provide an explicit characterization of the Nash equilibrium of Cournot for a specific demand function. Suppose that both firms have marginal cost c and the inverse demand function is given by P (Q) = α βq, where Q = q 1 + q 2, where α > c. Then player i will maximize: max π i (q 1, q 2 ) q i 0 = [P (Q) c] q i = [α β (q 1 + q 2 ) c] q i. To find the best response of firm i we just maximize this with respect to q i, which gives first-order condition [α c β (q 1 + q 2 )] βq i = 0. Therefore, the best response correspondence (function) of firm i can be written as α c βq i q i =. 2β 32

33 Examples Cournot Competition (continued) Now combining the two best response functions, we find the unique Cournot equilibrium as q 1 = q 2 = α c. 3β Total quantity is 2 (α c) /3β, and thus the equilibrium price is α + 2c P =. 3 It can be verified that if the two firms colluded, then they could increase joint profits by reducing total quantity to (α c) /2β and increasing price to (α + c) /2. 33

34 Examples Examples: Bertrand Competition An alternative to the Cournot model is the Bertrand model of oligopoly competition. In the Cournot model, firms choose quantities. In practice, choosing prices may be more reasonable. What happens if two producers of a homogeneous good charge different prices? Reasonable answer: everybody will purchase from the lower price firm. In this light, suppose that the demand function of the industry is given by Q (p) (so that at price p, consumers will purchase a total of Q (p) units). Suppose that two firms compete in this industry and they both have marginal cost equal to c > 0 (and can produce as many units as they wish at that marginal costs). 34

35 Examples Bertrand Competition (continued) Then the profit function of firm i can be written as Q (p i ) (p i c) if p i > p i 1 π i (p i, p i ) = 2 Q (p i ) (p i c) if p i = p i 0 if p i < p i Actually, the middle row is arbitrary, given by some ad hoc tiebreaking rule. Imposing such tie-breaking rules is often not kosher as the homework will show. Proposition In the two-player Bertrand game there exists a unique Nash equilibrium given by p 1 = p 2 = c. 35

36 Examples Bertrand Competition (continued) Proof: Method of finding a profitable deviation. Can p 1 c > p 2 be a Nash equilibrium? No because firm 2 is losing money and can increase profits by raising its price. Can p 1 = p 2 > c be a Nash equilibrium? No because either firm would have a profitable deviation, which would be to reduce their price by some small amount (from p 1 to p 1 ε). Can p 1 > p 2 > c be a Nash equilibrium? No because firm 1 would have a profitable deviation, to reduce its price to p 2 ε. Can p 1 > p 2 = c be a Nash equilibrium? No because firm 2 would have a profitable deviation, to increase its price to p 1 ε. Can p 1 = p 2 = c be a Nash equilibrium? Yes, because no profitable deviations. Both firms are making zero profits, and any deviation would lead to negative or zero profits. 36

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