Expected Payoff

Definition

Expected payoff is a calculated value that represents the average outcome of a decision or strategy in a game, factoring in the probabilities of different outcomes. It helps players determine the best strategy by weighing potential benefits against risks, making it essential for analyzing both pure and mixed strategies, as well as identifying Nash equilibria in strategic situations.

5 Must Know Facts For Your Next Test

1. Expected payoff is calculated by multiplying each possible outcome's value by its probability and summing those products.
2. In pure strategies, expected payoffs are often straightforward as players select one specific action without randomness.
3. Mixed strategies involve probabilities assigned to different actions, making expected payoffs more complex as players must account for various possible responses from opponents.
4. Finding mixed strategy Nash equilibria requires calculating expected payoffs for all players and ensuring that no player has an incentive to deviate from their assigned strategy.
5. Expected payoffs can vary based on the risk preferences of players; some may prefer higher risk for potentially higher rewards, while others may seek more stable, lower returns.

Review Questions

• How does expected payoff influence the choice between pure and mixed strategies in a game?
  • Expected payoff plays a crucial role in guiding players' decisions between pure and mixed strategies. In pure strategies, players make definite choices aiming for the highest expected payoff based on certain outcomes. In contrast, mixed strategies introduce randomness to balance potential outcomes against uncertainties, allowing players to optimize their expected payoffs by considering opponents' likely responses.
• Discuss how calculating expected payoffs is essential for determining mixed strategy Nash equilibria.
  • Calculating expected payoffs is vital when finding mixed strategy Nash equilibria because it allows players to assess their optimal strategies in response to the mixed strategies of others. Each player's expected payoff must be equal across different strategies to ensure that no one has an incentive to switch. This equilibrium condition emerges from carefully analyzing the probabilities and corresponding payoffs associated with various strategic combinations.
• Evaluate the impact of risk preferences on the concept of expected payoff and decision-making in strategic interactions.
  • Risk preferences significantly influence how players interpret expected payoff and make decisions in strategic interactions. Players who are risk-averse may favor strategies that guarantee more stable outcomes with lower expected payoffs rather than taking risks for potentially higher rewards. Conversely, risk-seeking players might gravitate towards strategies with high variability in outcomes, valuing the chance for larger payoffs even if they come with higher uncertainty. This dynamic shapes not only individual strategies but also the overall behavior of players within a game.

Related terms

Payoff Matrix: A table that displays the payoffs for each player based on the combination of strategies chosen in a game.

Nash Equilibrium: A situation in a game where no player can benefit from unilaterally changing their strategy, as all players' strategies are optimal given the strategies of others.

Probability Distribution: A function that shows all possible values of a random variable and their associated probabilities, crucial for calculating expected payoffs.