Information, Control and Games Shi-Chung Chang EE-II 245, Tel: ext Office Hours: Mon/Wed 1:00-2:00 pm or by appointment Yi-Nung Yang (03 ) ext. 5205,

Normal Form (one-shot) games. Solution concepts: the Nash Equilibrium. Lecture 2

What is a game? A finite set N of players –N = { 1, 2, …., i, ……n} –A set of strategies S i for each player Strategies (actions ) set –S i = { s i : s i is a strategy available to player i } S i may be finite or infinite. A payoff function  i for each player. –  i assigns a payoff to player i depending on which strategies the players have chosen.

Example 1 Working on a joint project – 兩人合作寫 term project Both work hard: One works hard but the other goofs off Both goof off

Example 1 (cont.) Working on a joint project: –A finite set N of players N = { 1, 2} A set of strategies S i for each player –Strategies (actions ) set S i = { work hard, goof off} –A payoff function  i for each player.  1 (W, W)= 2 =  2 (W,W)  1 (W, G)= 0 =  2 (G,W)  1 (G, W)= 3 =  2 (W,G)  1 (G, G)= 1 =  2 (G,G)

Example 1 (cont.) Normal Form

Example 2 Duopoly – 價格戰 : 中油 vs 台塑

Example 3 Coordination game Battle of the Sexes (BoS) – 合則兩利, 不合則兩害

Solution to The famous Prisoner’s Dilemma Prisoner’s Dilemma—an engineer’s version –Suppose each of two engineers wants to build a bridge or a tunnel across the Amazon from city A to city B. –It costs $20 million to build a bridge and $50 million to build a tunnel. –Revenue If both build a bridge/tunnel, each can sell her bridge/tunnel for $80 million. if one builds a bridge and one builds a tunnel, the bridge will sell for $25 million and the tunnel will sell for $120 million Why? Due to high winds and heavy rains in the area, most people when given a choice will choose to drive through a tunnel.

N = {1, 2} S 1 = {bridge, tunnel} = S 2  I (bridge, bridge) = 80-20=60=  2 (bridge, bridge)  I (tunnel, tunnel) = 80-50=30=  2 (tunnel, tunnel)  I (bridge, tunnel) = 25-20=5=  2 (tunnel, bridge)  I (tunnel, bridge) = =70=  2 (bridge, tunnel) x Bimatrix Form: bridge tunnel Player 2 bridge Player 1 60, 605, 70 70, 530, 30 tunnel 10

Solution Concepts What is a solution to a game? –We want a solution to predict what strategies players will choose. –Note: solutions can also be prescriptive—they can tell us what strategies players should play. We will concentrate for now on the predictive performance of a solution. –We can test a solution’s predictive ability experimentally, by having subjects (often students) play games in a laboratory or empirically, by seeing how firms behave in a market, or how politicians behave in an election.

The Premier solution concept: The Nash Equilibrium –We will use the PD game to introduce the concept. –Why is this the premier concept in game theory? Because it has performed relatively well in experimental tests and empirical tests and is widely applicable. (See Osborne’s discussion p.25).

Let’s look back at our bimatrix form of Prisoner’s Dilemma. –Our two engineers see the 60, 60 payoff and would like to agree to build bridges. –However, even if they meet to talk things over and agree- -- Engineer 1 will Reason as follows: If 2 builds a bridge I can earn 60 or defect to a tunnel and earn 70. And if 2 cheats and builds a tunnel, I will earn 5 or defect and earn 30. –So no matter what 2 does I do better building a tunnel!!! Therefore I build a tunnel –Player I reasons similarly and builds a tunnel. Each earns 30.

We have two solution concepts so far Players agree to jointly “optimize”: (bridge, bridge) this is also called a Pareto optimal outcome Dominant strategy equilibrium: (tunnel, tunnel) –Why would we predict that the dominant strategy will be played and not the joint optimal solution? –The logic to playing the dominant strategy equilibrium is overwhelming. I earn more with tunnel no matter what my opponent does. –Moreover, dominant strategy equilibrium tests well in experiments in the lab even in Prisoner’s Dilemma (see Osborne’s discussion). All sorts of examples too. OPEC, anti-trust cases (firms found to have cheated on price agreements, avoidance of PD etc)..

Pareto Optimal Outcome? When will the players play the joint optimal solution? –When it is possible for players to form legal binding commitments. For example, write a legal contract. Conclusion: –On the day of the press conference the two engineers announce what each will build. –They may have agreed before hand to build bridges. But they will both announce: TUNNEL –Unless they were able to write an enforceable contract. –Not so easy to do in most situations and often illegal. –Firms have to make their way around Prisoner’s Dilemma!

Comments: 1. We are discussing noncooperative game theory where no binding contracts are allowed. 2.We are discussing one-shot simultaneous play games where both players must announce their strategies simultaneously and the game is played once. 3.We have seen an example of a dominant strategy equilibrium. Here’s the definition for a two player game. A dominant strategy equilibrium is a strategy pair (s 1 *, s 2 * ) such that s 1 *  S 1, s 2 *  S 2,  I (s 1 *, s 2 ) >  I (s 1, s 2 ) for all s 1  S 1, s 2  S 2,and s 1 not equal to s 1 *  2 (s 1, s 2 * ) >  2 (s 1, s 2 ) for all s 1  S 1, s 2  S 2,and s 2 not equal to s 2 *. 4.Most games don’t have dominant strategy equilibrium. That’s why Nash introduced the Nash equilibrium, which generalizes the dominant strategy equilibrium with which it shares its defining characteristic, stability.

Definition A Nash equilibrium is an strategy profile s* with the property that no player i can do better by choosing and action different from s i *, given that every other player j adheres to s j * s* = {s i *, s j *}

Now let’s define a Nash equilibrium. We will look at a game that has a Nash equilibrium, but no dominant strategy Equilibrium. Here are four equivalent definitions of a Nash equilibrium. First two give us a feeling for what a Nash equilibrium is. The second two are useful for funding the Nash equilibrium or equilibria for a specific game.

Given a game G = (N = {1,2}; S 1, S 2 ;  I,  2 ), the strategy pair (s 1 *, s 2 * ) is a Nash equilibrium for G if 1.Neither player has an incentive to unilaterally defect to another strategy. 2. s 1 * is a best response to s 2 * and s 2 * is a best response to s 1 *. 3.  1 (s 1 *, s 2 * )   1 (s 1, s 2 * ) for all s 1  S 1. and  2 (s 1 *, s 2 * )   2 (s 1 *, s 2 ) for all s 2  S  I (s 1 *, s 2 * ) is a column maximum and  2 (s 1 *, s 2 * ) is a row maximum.

Uniqueness of Nash equil. If a Nash equil. exist, is it unique? Example: Battle of the sexes –It is a Saturday night, Geroge loves to watch football, but Marry enjoys opera.... –They also like each other’s company... –Find the Nsah equil.?

Existence of Nash equilibrium Same as before with a slight modification –George wants to meet Marry. However, Marry wants to avoid George –The only activities are a movie and a dance –Marry prefers to be alone, but if she must be with George, she prefers the movie, since she won’t have to talk to George. –George prefers to be with Marry, and if he succeeds, he prefers the dance, where he can talk to her. –Find Nash eq.

Exercise: A Sealed Bid Auction. Suppose two bidders bid for an item they know they can sell for $20. The rules of the auction require a bid of $16, $10, or $4. If both bidders submit the same bid, they share the item. Put the game in normal form.

1 2 H M L H M L 2,2 4,0 4,0 0,4 5,5 10, 0 0,4 0, 10 8,8

1. Is there a dominant strategy equilibrium? What is player 1’s best response to H? What is player 1’s best response to M? What is player 1’s best response to L? So no one strategy of player 1 is a best response to all strategies of player 2. 2.Find all Nash equilibria. Is (H,H) a Nash equilibrium? Is (H,M) a Nash equilibrium? etc. Note a Nash equilibrium is a strategy profile and should not be given in terms of payoffs. 所謂 Nash equil. 係指策略組合, 非報酬組合

Strict and nonstrict equilibria

Dominant Strategy For player 1: –T is dominated by M –M is dominated by B

Cournot Game Cournot’s duopoly game –Two firms produce identical products and competes in a market –Market demand: P(Q) = P(q 1 + q 2 ), P' (Q) < 0 –Each firm’s profit: Revenue - Cost  i (q i, q -i )= P(q i + q -i ) q i - C i (q i ) –Optimization: maximizing profits FOC:  i (q i, q -i )/  q i = P'(q i + q -i ) q i +P - C' i  0, for i=1, 2 –Best response function (reaction curve) q i = q i (q -i ) –Solve q 1, q 2 simultaneously to yield Nash solution

Cournot Game: an example Market Demand P=P(Q) =  - Q, Q= q 1 + q 2 Common Constant Marginal Cost C i = c q i, for i = 1,2 Profits  i (q i, q -i )= (  - q i - q -i )q i - cq i FOC: Response function  i (q i, q -i )/  q i = -q i +(  -q i - q -i -c)  0

Profit Function in Cournot Game Profit function:  1 = q 1 (  -q 1 - q 2 -c) given any q 2 when q 2 = 0  1 = q 1 (  -q 1 -c) q 1 = 0,  -c when q 2 > 0 Profit curve shifts downward  1 = q 1 (  -q 1 - q 2 -c)

Best Response f() and Nash Firm i’s optimal choice of q i given other’s q -i For firm 1, FOC becomes (-1)q 1 +(  -q 1 - q 2 -c)  0 q 1 = (1/2) (  -q 2 -c) For firm 2, FOC becomes -q 2 +(  -q 1 - q 2 -c)  0 q 2 = (1/2) (  -q 1 -c) Cournot-Nash equilibrium q i * = (1/3) (  -c), for i =1, 2

Reaction Curves and Nash

Nash Equil. In Cournot Game

A Collusive Duopoly Outcome Two firms collude as a monopoly –They maximize joint profits and share the output –Market Demand: P=P(Q) =  - Q, –Joint Profits max  = P(Q)Q - cQ = (  - Q)Q - cQ –FOC (  - Q) -Q - c =0 => Q m * =q 1 +q 2 =(  -c) /2  Each firm’s collusive output q i m * = (  -c) /4 < q i * = (  -c) /3 OPEC collusion

A Collusive Duopoly Outcome is not a Nash equilibrium?

Bertrand’s Competition Price (cost) competition –Firms set prices to maximize profits –Consumers purchase with the lowest price –A Firm takes ALL with the lowest price. Firms share the market equally if prices are the same The Game –Player: the firms (with cost function C i (q i ) –Strategies: each firm’s possible (non-negative) prices –Payoffs for firm i: (market demand D=  - p) p i D(p i ) / m - C i (D(p i )/m) if there are m firms with the same lowest price, where m = 1 if firm i’s profits is lower than the others

Profit function in Bertrand Game Bertrand’s duopoly game –Two firms compete in the market

Profits in Duopoly Bertrand when p j p j –Best response B i (p j )={p i : p i >p j } when p j = c, similar to the above profit =0 if p i  p j

Profits in Duopoly Bertrand (2) when c p j –Best response seems to be empty set

Profits in Duopoly Bertrand (3) when p j > p m, firm i’s best response B i (p j )={p i : p i =p m }

Best Response f() in Bertrand

Best Response Plot in Bertrand Nash equilibrium: (p 1 *, p 2 *) = (c, c)

Reasoning in Bertrand No one should set p i < c since profit<0 so, feasible strategy set is {p i  c}, for i=1,2 If firm i choose p i < p j, firm j can further lower p j to take All market. But firm i also does the same thing. So the price continued to be lower (price war) until p i = c. Zero-profit Nash outcome –zero profit => normal profit