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Lecture 52 Analytical Game Theory

Prisoners' Dilemma

  • Two criminals enough evidence to arrest on a minor charge. Two separate crimes
  • Minor = 2 years in jail (if they cooperate \(C\))
  • Defect \(D\)
    • Defector goes free
    • Cooperator 20 years in jail
    • Both defect: 10 years (as a reward for testifying)

Each prisoner cares about nothing but the number of years he himself spends in jail.

\[ \begin{matrix} & \text{Cooperate} & \text{Defect} \\ \text{Cooperate} & (-2,-2) & (-20,0) \\ \text{Defect} & (0, -20) & (-10,-10) \end{matrix} \]
  • Dominant strategy: \(X\) > \(Y\) if choosing \(X\) is better than choosing \(Y\) no matter what the other player does.
    • For each player, defecting is the better strategy, if you compare 0 or -10 (while other player selects any).
    • No rational player will cooperate.
  • Pareto dominate outcome: \(X\) > \(Y\) if all players weakly prefer \(X\) to \(Y\) and at least one player strictly prefers \(X\) to \(Y\).

  • Pareto optimal outcome: if it is not pareto dominated by any other outcome.

  • \((C,C)\) pareto dominates \((D,D)\)… so Adam Smith's statement is refuted1

Game with No NE in Pure Strategy

  • Want to study in library or dept but don't want to run into bestie
  • Bestie however wants to study with you

1 if his/her goal is reached. 0 otherwise

  • Player I: You
  • Player II: Bestie
\[ \begin{matrix} & \text{Library} & \text{Department} \\ \text{Library} & (0,1) & (1,0) \\ \text{Department} & (1,0) & (0,1) \end{matrix} \]
  • No NE in this game.

  • Also known as matching pennies

    • 2 players flip 2 coins.
    • Both same \(\implies\) Player 1 wins
    • Both different \(\implies\) Player 2 wins
    • (In the library-department game, the order is reversed… but get the point.)

  • Zero sum game

    • Gains and losses of individuals sum to zero for all combinations of strategies they might pursue.

Nash Equilibrium in mixed strategies

  • Some games have no NE in pure strategies.
  • They might have NE in mixed strategies.
  • Suppose in the study example the choice is made based on a fair coin flip. So the chances of either person ending up somewhere is 50-50. Thus, you can't do any better to decide. Hence be indifferent between library and department.
  • This way, both are in NE though we are playing mixed strategies. But mixed-strategy equilibrium is tricky to find in general.

For the battle of the sexes:

\[ \begin{matrix} & \text{Concert}(q) & \text{Theatre}(1-q) \\ \text{Concert}(p) & (2,1) & (0,0) \\ \text{Theatre}(1-p) & (0,0) & (1,2) \end{matrix} \]
  • The player must be indifferent between the two strategies (they themselves don't tell us anything)
  • So, \(u(\text{Concert})=u(\text{Theatre})\)
  • For player I (she, concert)
    • \(E(u(\text{Concert})) = 2q+0\times(1-q)\)
    • \(E(u(\text{Theatre})) = 0\times q+1\times(1-q)\)
    • So, \(2q = 1-q\) and thus, \(q=\dfrac{1}{3}\)
  • For player II,
    • \(p = \dfrac{2}{3}\) (similarly calculated)

So

There is a NE in mixed strategies in which P\(I\) selects concert with probability \(\dfrac{2}{3}\) and P\(II\) with probability \(\dfrac{1}{3}\).

In this mixed-strategy equilibrium
- Player I, \(u(C) = u(T) = 2q = \dfrac{2}{3}\)
- Player II, \(u(C) = u(T) = p = \dfrac{2}{3}\)

Examples

\[ \begin{matrix} & \text{L} & \text{R} \\ \text{U} & (5,2) & (1,1) \\ \text{D} & (1,1) & (2,5) \end{matrix} \]
  • Pure: (UL) and \((D,R)\)
  • Mixed: \(q=\dfrac{1}{5}, p=\dfrac{4}{5}\)

So basically, we are finding probabilities for which the utilities would be equal, i.e. they will be indifferent.

\[ \begin{matrix} & \text{L} & \text{R} \\ \text{U} & (1,1) & (0,0) \\ \text{D} & (0,0) & (0,0) \end{matrix} \]
  • Pure: \((U,L)\) and \((D,R)\)
  • Mixed: none (as most of the payoffs are zero)

Pure vs. Mixed Equilibria

  • Don't use your cross court shot every time or the opponent will learn to expect it
  • Thus we must mix it up a bit… hit your weaker short… to keep the opponent guessing

Playing the inferior option from time to time doesn't mean NOT playing an equilibrium strategy

Spousonomics

  • If you are stuck in an equilibrium where you do the dishes, make the bed and chores, while your spouse sits back and relaxes
  • change your spouse into an acceptable partner by playing a mixed strategy…
  • sometimes and sometimes not
    • do the laundry
    • make the bed
    • etc

Chess

  • Finite game = finite number of moves t o choose from at any point in the game.
  • Finite game \(\implies\) Nash's theorem says it has an equilibrium. (Only existence, not what they are)

\(\implies\) Chess should be uninteresting between experienced players (both play the equilibrium strategy… like tic-tac-toe)

Nash's Theorem

  • Every finite game has a Nash equilibrium
    • Finite game = in which all players have finite number of pure strategies.
  • So, if it is easy to check if the Nash Theorem condition is satisfied, we know that at least one NE in Pure or mixed strategies exist!

  1. He said that rational pursuit of individual's self-interest leads to socially desirable outcomes. But it doesn't always. ↩