Game Theory Basics The prisoners dilemma in real
Game Theory Basics
The prisoner’s dilemma in real life • The American / Soviet arms race during the Cold War. • Should the US disarm? Does it matter what the Soviets plan is?
Can we build a payoff matrix? Soviet Union Arm Disarm Arm US Disarm
Can we build a payoff matrix? Soviet Union Arm Disarm 1 Arm +2 1 US -2 0 +2 Disarm -2 0
Rationale behind payoffs • Both countries disarm = less military spending, less chance of accidental attack, no military superiority • Both countries arm = more military spending, more dangerous world • One country disarms, but not the other = severe military mismatch!
Dominate strategy for both: Not knowing what the other country will do, you choose to be armed. Soviet Union Arm Disarm Dominate strategy 1 Arm +2 1 US -2 0 +2 Disarm -2 0
Nash equilibrium: Neither player would change their strategy, even if they knew in advance what the other player would do. Soviet Union Arm Disarm Nash Equilibrium 1 Arm +2 1 US -2 0 +2 Disarm -2 0
Example #2 • Imagine the following scenario: Your teacher assigns a project, and you are to work with a partner. You would like an A, but you don’t want to do all the work. • Imagine the following payoffs:
Joint project payoffs • Both work hard; get a A (A=40) • Only one works hard, get a B (B=30) • Neither one works hard, get a D (D=10) • The cost of hard work = 25 (After all, you could be having fun with your friends!) • Can you build a payoff matrix? ? ?
Homework project payoff matrix Your decision Work Slack Work Classmate Slack
Homework project payoff matrix Your decision Work Slack 40 -25=15 Work 40 -25=15 Classmate Slack 30 -0=30 30 -25=5 30 -0=30 10 -0=10
Homework project payoff matrix Your decision Work Slack 40 -25=15 Work 40 -25=15 Classmate Slack 30 -0=30 30 -25=5 30 -0=30 10 -0=10 Dominate strategy = slack!
Example #3 Think of rock, paper, scissors. • In a two person game, is there a dominant strategy? • Is there a Nash Equilibrium?
Rock, paper, scissors payoffs Player 1 P R R Player 2 P S 0 0 1 -1 -1 -1 1 0 1 S 1 0 -1 1 -1 0 -1 1 0
Is there a dominant strategy? Player 1 P R R Player 2 P S 0 0 1 -1 -1 -1 1 0 1 S 1 0 -1 1 -1 0 -1 1 0
If each player chooses R, P, or S randomly (exactly 1/3 of the time) there is no dominant strategy. Player 1 R P S R Player 2 P S 0 0 1 -1 -1 -1 1 0 -1 1 -1 0 -1 1 0
There is no Nash equilibrium; if you knew what the other player was going to do, you might switch. Player 1 R P S R Player 2 P S 0 0 1 -1 -1 -1 1 0 -1 1 -1 0 -1 1 0
You would only have a dominant strategy if you knew what the other player was likely to do. A bad player might have a tendency to make one of the choices more often, or in some predictable pattern. Player 1 P R R Player 2 P S 0 0 1 -1 -1 -1 1 0 1 S 1 0 -1 1 -1 0 -1 1 0
Example #4 The only way to get to Mackinaw Island is by ferry boat. • Assume there are only two ferry lines, the red fleet and the black fleet. (duopoly) • Each company has to decide whether to charge a low price or a high price.
Assume the following payoff matrix per passenger Red fleet low price high price $2 Low price Black fleet $0 $5 $2 $4 $5 High price $0 $4
Questions: Is there a dominant strategy for the red fleet? For the Black fleet? Red fleet low price high price $2 Low price Black fleet $0 $5 $2 $4 $5 High price $0 $4
Questions: Is there a Nash equilibrium? Red fleet low price high price $2 Low price Black fleet $0 $5 $2 $4 $5 High price $0 $4
Dominant strategy equilibrium Nash equilibrium Red fleet low price high price $2 Low price Black fleet $0 $5 $2 $4 $5 High price $0 $4
Repeated games • In real life, this wouldn’t be a one-shot game. The ferry boats would be competing against each other all summer, or for years to come. Both companies would be better off if they charged the higher price. A tit for tat strategy could allow tacit collusion to occur. • (Tacit collusion = more profits for both)
Main points • The prisoner’s dilemma can appear in many different situations • Cooperating with the other player could lead to a higher payoff than simply following the dominant strategy • In oligopoly, producers who can cooperate can achieve monopoly pricing power
Is collusion new? “People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices” Adam Smith The Wealth of Nations 1776
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