Information, Control and Games

Information, Control and Games Shi-Chung Chang EE-II 245, Tel: 2363-5251 ext. 245 scchang@cc.ee.ntu.edu.tw, http://recipe.ee.ntu.edu.tw/scc.htm Office Hours: Mon/Wed 1:00-2:00 pm or by appointment Yi-Nung Yang (03 ) 2655201 ext. 5205, yinung@cycu.edu.tw

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 Si for each player • Strategies (actions ) set • Si = { si : si is a strategy available to player i } Si may be finite or infinite. • A payoff function i for each player. • iassigns 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 Si for each player • Strategies (actions ) set • Si = { 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 gameBattle 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.

Player 2 bridge tunnel bridge 60, 60 5, 70 Player 1 tunnel 70, 5 30, 30 N = {1, 2} S1 = {bridge, tunnel} = S2 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) = 120-50=70= 2 (bridge, tunnel)x Bimatrix Form: 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 noncooperativegame theory where no binding contracts are • allowed. • We are discussing one-shot simultaneous play games where both players must • announce their strategies simultaneously and the game is played once. • 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 (s1*, s2*) such that s1*  S1, • s2*  S2 , • I (s1*, s2) > I (s1, s2) for all s1 S1, s2 S2 ,and s1 not equal to s1* • 2 (s1, s2*) > 2(s1, s2) for all s1 S1, s2 S2 ,and s2 not equal to s2*. • 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 si*, given that every other player j adheres to sj*s* = {si*, sj*}

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 Nashequilibrium. 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}; S1, S2 ; I, 2), the strategy pair (s1*, s2*) is a Nash equilibrium for G if 1.Neither player has an incentive to unilaterally defect to another strategy. 2. s1* is a best response to s2*and s2*is a best response to s1*. 3. 1 (s1*, s2*)  1 (s1, s2*) for all s1 S1. and 2 (s1*, s2*)  2 (s1*, s2) for all s2 S2. 4. I (s1*, s2*) is a column maximum and 2 (s1*, s2*) 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.

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

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. • 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 • 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(q1+ q2), P' (Q) < 0 • Each firm’s profit: Revenue - Costi (qi, q-i)= P(qi+ q-i) qi - Ci(qi) • Optimization: maximizing profitsFOC: i(qi, q-i)/qi = P'(qi+ q-i) qi +P - C'i 0, for i=1, 2 • Best response function (reaction curve)qi = qi(q-i) • Solve q1, q2 simultaneously to yield Nash solution

Cournot Game: an example • Market DemandP=P(Q) =  - Q, Q= q1+ q2 • Common Constant Marginal CostCi = c qi , for i = 1,2 • Profitsi (qi, q-i)= ( - qi- q-i)qi - cqi • FOC:Response functioni (qi, q-i)/qi = -qi+( -qi- q-i -c)  0

Profit Function in Cournot Game • Profit function:1=q1( -q1- q2 -c)given any q2when q2 = 01= q1( -q1-c) q1 = 0, -cwhen q2 > 0Profit curve shifts downward1= q1( -q1 - q2 -c)

Best Response f() and Nash • Firm i’s optimal choice of qi given other’s q-i • For firm 1, FOC becomes(-1)q1+( -q1- q2 -c)  0q1 = (1/2) ( -q2 -c) • For firm 2, FOC becomes -q2+( -q1- q2 -c)  0q2 = (1/2) ( -q1 -c) • Cournot-Nash equilibriumqi* = (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 => Qm* =q1+q2=(-c) /2 Each firm’s collusive output qim* = (-c) /4 < qi* = (-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 Ci(qi) • Strategies: each firm’s possible (non-negative) prices • Payoffs for firm i: (market demand D=  - p)piD(pi) / m - Ci(D(pi)/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 pj < c, firm i’s profit <0 if pipj profit =0 if pi>pj • Best responseBi(pj)={pi: pi>pj} • when pj = c, similar to the aboveprofit =0 if pipj

Profits in Duopoly Bertrand (2) • when c < pj pm, firm i’s profit ↑in pi if pi<pj profit = 1/2 share if pi=pj profit =0 if pi>pj • Best response seems to beempty set

Profits in Duopoly Bertrand (3) • when pj > pm, firm i’s best responseBi(pj)={pi: pi=pm}

Best Response f() in Bertrand

Best Response Plot in Bertrand • Nash equilibrium: (p1*, p2*) = (c, c)

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

Information, Control and Games