Question

Here is a description of the simplest poker game. There are two players and only two cards in the deck, an Ace (A) and a King (K). First, the deck is shuffled and one of the two cards is dealt to player 1 . That is, nature chooses the card for player 1 . It is the Ace with probability \(1 / 2\) and the King with probability \(1 / 2\). Player 2 does not receive a card. Player 1 observes his card and then chooses whether to bid (B) or fold (F). If he folds, then the game ends with player 1 getting a payoff of \(-1\) and player 2 getting a payoff of 1 (that is, player 1 loses his ante to player 2 ). If player 1 bids, then player 2 must decide whether to bid or fold. When player 2 makes this decision, she knows that player 1 bid, but she has not observed player 1's card. The game ends after player 2's action. If player 2 folds, then the payoff vector is \((1,-1)\), meaning player 1 gets 1 and player 2 gets \(-1\). If player 2 bids, then the payoff depends on player 1 's card; if player 1 holds the Ace, then the payoff vector is \((2,-2)\); if player 1 holds the King, then the payoff vector is \((-2,2)\). Represent this game in the extensive form and in the Bayesian normal form.

Short answer

Game tree shows actions; Bayesian table shows strategies with Player 2's information asymmetry.

Step-by-step solution

Define Players and Actions

This poker game involves two players: Player 1 and Player 2. Each player can take specific actions. Player 1 can either bid (B) or fold (F) upon seeing their card. Player 2 can choose to bid or fold after observing Player 1's action. However, Player 2 does not see Player 1's card.

Determine Nature's Role

Nature plays a role at the beginning by randomly dealing either an Ace (A) or a King (K) to Player 1, each with a probability of \(\frac{1}{2}\). This decision by nature initiates the game and affects subsequent outcomes.

Illustrate Extensive Form of the Game

The extensive form is represented as a game tree: 1. The root node is Nature's move, where it picks either Ace (A) or King (K). 2. Player 1 observes the card and, based on it, chooses to bid (B) or fold (F). - If Player 1 folds (F), the game ends with payoffs (-1, 1). 3. If Player 1 bids (B), Player 2 then decides whether to bid or fold without knowing Player 1's card. - If Player 2 folds, the game ends with payoffs (1, -1). - If Player 2 bids, payoffs depend on Player 1’s card: - For Ace: (2, -2), - For King: (-2, 2).

Construct Bayesian Normal Form

The Bayesian normal form table represents strategies and payoffs considering Player 2’s uncertainty about Player 1’s card. It lists possible types of Player 1 (Ace or King), and strategies are combinations of decisions for bidding or folding: - Player 1's strategies: Choose separately for Ace and King, i.e., (B, B), (B, F), (F, B), (F, F). - Player 2’s strategy matrix consists of deciding B or F after Player 1 bids with unknown card type. - Expected payoffs are calculated based on the probabilities of Player 1's card being Ace or King and Player 2 responding.

Calculate Expected Payoffs

For determining expected payoffs, consider the likelihood of each scenario: - If Player 1 is dealt an Ace and chooses B, while Player 2 chooses to bid, payoff is (2, -2). - With a King and Player 2 bids, payoff is (-2, 2). - Aggregate these based on the probability of Player 1 holding either card and Player 2's responses to form the final matrix representing Bayesian normal form.

Key concepts

Extensive Form Representation

In game theory, extensive form representation is a way of illustrating a strategic situation where decisions are made at different points in time. In this poker game, the extensive form can be visualized using a game tree, which clearly outlines each player's choices and possible outcomes.
The game begins with nature's move, where it randomly deals either an Ace (A) or a King (K) to Player 1. This decision by nature is crucial in shaping the rest of the game since it affects Player 1's strategy.
Next, Player 1 observes their card and must decide whether to bid (B) or fold (F). If Player 1 folds, this ends the game immediately, resulting in a payoff of (-1, 1), meaning Player 1 loses their bet and Player 2 gains.
Should Player 1 bid instead, Player 2 then makes a decision to bid or fold without the knowledge of Player 1's card. If Player 2 folds, Player 1 wins the game with a payoff of (1, -1).
If Player 2 decides to bid, the outcome depends entirely on Player 1's card. For an Ace, the payoffs are (2, -2); with a King, the payoffs turn to (-2, 2). This structure effectively captures the sequence of decisions and the strategic dependency on the information revealed through the game.

Bayesian Normal Form

The Bayesian normal form is used to represent games with incomplete information. In this poker scenario, Player 2 does not know the card dealt to Player 1. Hence, the uncertainty is modeled in the game's representation.
The Bayesian normal form accounts for the different types of Player 1's hand (Ace or King) and provides a way to specify strategies for each possible situation.
Player 1 has four possible strategies depending on the card they hold:

• Bid with both Ace and King (B, B)
• Bid with Ace, fold with King (B, F)
• Fold with Ace, bid with King (F, B)
• Fold with both Ace and King (F, F)

Player 2's strategy involves deciding to bid or fold upon Player 1's action of bidding, without knowing if Player 1 holds an Ace or a King. This uncertainty is at the core of Bayesian analysis.
The Bayesian normal form, therefore, incorporates this element of chance and unknown information by employing a strategy matrix that considers these uncertainties and different scenarios.

Poker Game Strategy

Strategy in poker, even in its simplest form, is about making decisions that maximize a player's expected payoff based on available information. For Player 1, strategy involves deciding when to bid or fold based on the card received.
When Player 1 receives an Ace, bidding typically offers a strategic advantage due to the higher payoff potential if Player 2 also bids. In contrast, folding with an Ace ensures a small loss, which might not be optimal given the circumstances.
If a King is dealt, the decision to bid or fold is more nuanced. Bidding can be used as a bluff to sway Player 2's action, attempting to induce a fold for a potential payoff of (1, -1). On the other hand, folding immediately will reduce losses.
Player 2's strategy revolves around interpreting Player 1's bid without knowing the card. If Player 2 perceives Player 1 may be bluffing with a King, they might opt to bid, expecting a payoff of (-2, 2) if their suspicion holds true.
However, the risk of misjudging Player 1's position could lead to a significant loss. As such, the decision to bid or fold involves thoughtful consideration of probabilities, payoffs, and potential bluffs.

Expected Payoff Calculation

Expected payoffs help a player assess the likely benefit or loss from a particular strategy over time. This is crucial in decision-making, especially in games involving uncertainty like poker.
To calculate expected payoffs in this game, both the probability of events and the potential payoffs must be considered. With Player 1 holding an Ace, the probability is \(\frac{1}{2}\). If Player 1 bids and Player 2 also bids, the payoff becomes (2, -2).
In contrast, if Player 1 has a King, again with a probability of \(\frac{1}{2}\), and both players choose to bid, the outcome changes to (-2, 2).
The expected payoff for each player's strategy is determined by multiplying the probabilities by their respective payoffs and summing them up.

• For Player 1: \(expected \, payoff = \frac{1}{2} \times 2 + \frac{1}{2} \times (-2)\)
• For Player 2: \(expected \, payoff = \frac{1}{2} \times (-2) + \frac{1}{2} \times 2\)

These calculations guide the players in choosing strategies that yield the highest or most balanced expected gains, contributing to the broader strategy discussion.