Game theory is increasingly relevant in reinforcement learning where we have multiple agents. Understand the concept of Nash Equilibrium.
Game Theory

Game Theory

  • Mathematics of conflict.
  • Multiple agents.
  • Used in economics.
  • Increasingly a part of AI/ML.

1. Two-player zero-sum finite deterministic game of perfect information

    • A's points: leaves.
    • B's points: negative of A.
      • Pure strategy: 2
        • A: maximize
        • B: minimize

Minimax $\equiv$ Maximax Strategy

  • Both A and B would consider the worst case counter strategy.
    • A: maximize points.
    • B: minimize points.
  • There would always have an optimal pure strategy for each player.
    • You are assuming they are doing the same thing.
    • You are also assuming they are assuming everyone else are doing the same thing.

2. Two-player zero-sum finite non-deterministic game of perfect information

    • Pure Strategy: -2
      • A would maximize
      • B would minimize

3. Two-player zero-sum finite non-deterministic game of hidden information

  • Mini-poker
    • A is dealt a card, red or black with a probability of 50% each
      • Red is bad for A
      • Black is good for A
    • A may resign if given a red card: - 20 for A
      • Else A can hold:
        • B resigns: + 10
        • B sees:
          • If red: - 40
          • If black: + 30
  • Minimax is not equal to Maximax in this scenario.
    • A's strategy depends on what B would do.
    • B's strategy depends on what A would do.
    • We can use a mixed strategy here.
      • There's a distribution over strategies.
  • Assume p is the probability of being a "holder" and A has a mixed strategy.
    • B: resigner
      • A's expected profit:
        • 10p + (1-p)(-5) = 15p - 5
    • B: seer
      • A's expected profit:
        • -5p + (1-p)(5) = -10p + 5
    • We equate the equations to find out that we have p=0.4, where the expected value of game is +1.

4. Two-player non-zero-sum finite non-deterministic game of hidden information

  • Typical scenario where the district attorney offers a deal to both criminals to turn on one another.
  • No matter the response of the other prisoner, both would choose to defect. There is a dominance of a particularly strategy here.
  • This is also known as the Prisoner's Dilemma.

Nash Equilibrium

  • n players with strategies $S_1, S_2 ... S_n$
  • $S^*_1 \in S_1 ... S^*_n \in S_n$ are in Nash Equilibrium if
    • $\forall_i \ \ S^*_i = argmax_{s_i} utility_i(S^*_1 ... S^*_n)$ there is no reason for anyone to change.

Nash Equilibrium Notes

  • In the n-player pure strategy game, if elimination of strictly dominated strategies elimates all but one combination, that combination is the unique NE.
  • Any NE will survive elimination of strictly dominated strategies.
  • If n is finite and for all strategies which are finite, then there exist at least one NE.
  • If you have n-repeated game, you have n-repeated NE.