Notes on Uncertainty and Game Theory

Introduction to Uncertainty and Game Theory

  • Focus on concepts of uncertainty and game theory in economics.

Uncertainty

  • Definition: A situation where the outcomes or payoffs are not guaranteed.
  • Modeling Uncertainty:
    • Through random variables (probabilities/distributions).
    • Two states:
    • Certainty: Utility is straightforward.
    • Uncertainty: Expected utility used to evaluate choices.
Expected Utility Theory
  • Originated by Daniel Bernoulli and formalized by von Neumann-Morgenstern (VNM).
  • Key Points:
    • People care about the utility from outcomes rather than the outcomes themselves.
    • The expected utility calculation involves weighing outcomes by their probabilities:
      U(L)=U(xi)=π<em>iU(x</em>n)+(1−π<em>i)U(x</em>1)U(L) = U(xi) = \pi<em>i U(x</em>n) + (1 - \pi<em>i) U(x</em>1) where ( xn ) and ( x1 ) are outcomes of a gamble.
St. Petersburg Paradox
  • Demonstrates the discrepancy between expected value and actual utility.
    • Gamble example: Flipping a coin until heads appears. The payout doubles each round, leading to an infinite expected payoff, yet individuals do not value it as such.
Risk Aversion
  • Definition: Individuals prefer to avoid risk, valuing certainty over gambles with equal expected value.
  • Diminishing Marginal Utility: The additional utility gained from wealth decreases as wealth increases. Example:
    • An increase from $40,000 to $50,000 greatly improves wellbeing, but an increase from $50,000 to $60,000 provides less improvement.
Measurement of Risk Aversion
  • Absolute Risk Aversion (Pratt): The measure relates to the second derivative of the utility function:
    r(W)=−U<em>xx(W)U</em>x(W)r(W) = -\frac{U<em>{xx}(W)}{U</em>x(W)}
  • Relative Risk Aversion:
    r<em>r(W)=Wimesr(W)=−WU</em>xx(W)Ux(W)r<em>r(W) = W imes r(W) = -W \frac{U</em>{xx}(W)}{U_x(W)}

Game Theory

  • Definition: The study of strategic interactions among rational decision-makers.
  • Components of a game:
    1. Set of players ( N = {1, 2, …, n} )
    2. Each player's set of actions ( A_i )
    3. Strategies based on actions of players ( S_i )
    4. Outcomes leading to payoffs ( P_i )
    5. Common knowledge of rules of the game
    6. Representation as ( G = {S, P, A, I, N} )
Classifying Games
  • Categories include:
    • Timing of moves (simultaneously vs sequentially)
    • Information availability (perfect vs imperfect)
    • Type of actions (discrete vs continuous)
    • Game duration (one-shot vs repeated)
Solving Games
  • Normal Form: Representation of simultaneous-move games solved via Nash Equilibrium (NE).
  • Extensive Form: Representation of sequential-move games, solved via backward induction and Subgame Perfect Nash Equilibrium (SPNE).
Prisoner's Dilemma Example


  • Players choose simultaneously whether to cooperate or defect. The payoffs depend on both players' choices.

FinkSilent
Fink(1, 1)(3, 0)
Silent(0, 3)(2, 2)
Other Game Theory Topics
  • Mixed strategies and repeated games.
  • Games with incomplete information and signaling.
  • Experimental designs and evolutionary implications in game behavior.

Conclusion

  • Understanding uncertainty and game theory is crucial for analyzing economic behaviors and decision-making processes related to risk and strategy.