Question

Consider a simple simultaneous-bid poker game. First, nature selects numbers $x_1$ and $x_2$. Assume that these numbers are independently and uniformly distributed between 0 and 1. Player 1 observes $x_1$ and player 2 observes $x_2$, but neither player observes the number given to the other player. Simultaneously and independently, the players choose either to fold or to bid. If both players fold, then they both get the payoff $-1$. If only one player folds, then he obtains $-1$ while the other player gets 1 . If both players elected to bid, then each player receives 2 if his number is at least as large as the other player's number; otherwise, he gets $-2$. Compute the Bayesian Nash equilibrium of this game. (Hint: Look for a symmetric equilibrium in which a player bids if and only if his number is greater than some constant $\alpha$. Your analysis will reveal the equilibrium value of $\alpha$.)

Short answer

The Bayesian Nash equilibrium is for each player to bid if their number is greater than $4/9$.

Step-by-step solution

Understand the Payoffs in the Game

In this simultaneous-bid poker game, each player can either fold or bid. The payoff matrix is as follows: if both fold, each gets $-1$; if one folds and the other bids, the bidder gets $1$ and the folder gets $-1$; if both bid, they each get $2$ if their number is higher or equal to their opponent's number, otherwise $-2$.

Define the Strategy

We consider a symmetric Bayesian Nash equilibrium strategy where each player bids if their number $x_i$ is greater than a certain threshold $\alpha$. We need to determine this value $\alpha$.

Calculate Expected Payoffs

For each player observing $x_i$ and following the strategy 'bid if $x_i>\alpha$', we must calculate expected payoffs. If player 1 bids, with $x_1>\alpha$, player 1's payoff depends on whether $x_1$ is greater or lesser than $x_2$. If $x_1=x_2$, player 1 receives $2$. If $x_1>x_2$, player 1 also receives $2$; if $x_1<x_2$, they receive $-2$.

Calculate the Expected Payoff for Folding

If a player folds, their payoff is always $-1$ regardless of the value of $x_i$ because no other actions are taken by the player.

Equilibrium Condition

In equilibrium, the expected payoff from bidding should equal the payoff from folding $-1$. Therefore, calculate $\mathbb{E}[u(bid)]=2P(x_i\ge x_{-i}|x_i>\alpha )-2P(x_i<x_{-i}|x_i>\alpha )$. Using uniform distribution properties, the decision to bid will result in $\alpha =4/9$ as it makes the folding and bidding payoff equal.

Determine Value of $\alpha$

With $\mathbb{E}[u(bid)]=-1$, solve for $\alpha$. Integrating the expectations $P(x_i\ge x_{-i})={\int }_{\alpha }^1(1-x)dx$, which is the area where player 1 bids and wins, and $P(x_i<x_{-i})={\int }_{\alpha }^{1}xdx$, which is the area where player 1 bids and loses, yields $\alpha =4/9$.

Key concepts

Game Theory

Game theory is a fascinating mathematical framework designed to analyze strategic interactions among rational decision-makers. In our simultaneous-bid poker game, game theory helps us understand the strategic choices each player might make to maximize their payoffs—whether to fold or to bid. At the core of game theory, players aim to find their best-response strategies, considering the possible actions of others. In this scenario, each player's choice depends on predictions about the other player's actions, considering the lack of knowledge about their opponent's number.

• This poker game represents a class of games where players make decisions based on probabilities and potential outcomes.
• The essence of game theory lies in predicting opponent actions and maximizing potential gains.
• Finding the equilibrium in such games often leads to solutions referred to as Nash Equilibria, where each player has adopted the best strategy given the opponents' strategies.

Game theory's application in this poker scenario captures the crux of competitive strategies under uncertainty.

Simultaneous-move Games

Simultaneous-move games are a category of strategic games where all players make their decisions at the same time, without knowing the others' actions. This is exactly the situation our poker players face—they decide to fold or bid simultaneously. The lack of information about the opponent's choice adds a layer of complexity to these games.

• In simultaneous games, timing is crucial because decisions are made at the same instant.
• Players must rely on expectations and forecasts about others' decisions.
• These games typically do not allow for reconsideration or retraction of moves.

This structure requires players to carefully weigh their options and anticipate their opponents' choices, emphasizing the importance of strategic depth in decision-making.

Decision Making Under Uncertainty

Decision making under uncertainty is a critical skill in strategic games like our poker example. Here, each player lacks full information about the other's hand or intentions, making the decision process inherently uncertain. This uncertainty is due to the imperfect information—players know their number but not the opponent's.

• Players need to gauge the risk and reward of each possible choice under uncertain conditions.
• Decision-making strategies often rely on probabilities and payoff expectations.
• Risk management becomes essential where payoffs can vary significantly, as seen in the possible outcomes from folding or bidding.

In practice, players might use thresholds like $\alpha$ to decide their actions, optimizing based on expected outcomes weighed against the risks of uncertainty.

Payoff Matrix Analysis

Payoff matrix analysis allows players to visualize all possible outcomes and their respective payoffs, given the choices they and their opponents might make. In our poker game, the matrix helps both players contemplate the implications of folding or bidding. It considers all potential outcomes based on combinations of actions, quickly summarizing who wins what based on their strategies.

• The matrix outlines scenarios: both fold, one folds and one bids, or both bid, offering payouts for each.
• This helps players understand the potential rewards and penalties tied to each decision.
• Using this analysis, players can strategically decide whether to pursue aggressive or conservative strategies.

Analyzing with a payoff matrix can illuminate inherent risks and probable results, assisting players in calculating the most favorable path to follow based on expected payoffs.