Question

Draw the extensive-form representation of the following three-card poker game. There are three cards in the deck: an Ace, a King, and a Queen. The deck is shuffled, one card is dealt to player 1 , and one card is dealt to player 2 . Assume the deck is perfectly shuffled so that the six different ways in which the cards may be dealt are equally likely. Each player observes his own card but not the card dealt to the other player. Player 1 then decides whether to fold or bid. If he folds, then the game ends and the payoff vector is \((-1,1)\); that is, player 1 obtains \(-1\) (he loses his ante), and player 2 obtains 1 (gaining player 1 's ante). If player 1 bids, then player 2 must decide whether to fold or bid. If player 2 folds, then the game ends with payoff vector \((1,-1)\). If player 2 decides to bid, then the players reveal their cards. The player with the higher card wins, obtaining a payoff of 2 ; the player with the lower card loses, getting a payoff of \(-2\). Note that when both players bid, the stakes are raised. By the way, the Ace is considered the highest card, whereas the Queen is the lowest card.

Short answer

Extensive-form game tree constructed with decision nodes and payoff outcomes.

Step-by-step solution

Define the Game

We start by defining the elements of the three-card poker game. We have three cards: Ace (A), King (K), and Queen (Q). There are two players: Player 1 and Player 2. The deck is perfectly shuffled so that each player receives one of the three cards, leaving one card undistributed.

List All Possible Deal Combinations

There are 3 unique cards, so the deck can be dealt to Player 1 and Player 2 in 6 different combinations: (A,K), (A,Q), (K,A), (K,Q), (Q,A), and (Q,K). Each combination is equally likely.

Initial Decision by Player 1

For each card combination, Player 1 observes his card and makes the first decision: either Fold or Bid. Folding results in the payoff (-1, 1), ending the game. Bidding leads to Player 2 deciding on the next move.

Decision by Player 2 After Player 1 Bids

If Player 1 bids, Player 2 then decides whether to Fold or Bid. Folding results in the payoff (1,-1). Bidding means both players reveal their cards.

Resolution After Both Players Bid

When both players bid and show their cards, the player with the higher card wins. The payoff depends on the card rankings: Ace > King > Queen. The winning player receives a payoff of 2, and the losing player gets -2.

Constructing the Extensive Form Game Tree

Create the extensive-form game tree with the following nodes: - The root starts with nature deciding one of the 6 possible card combinations dealt to Player 1 and Player 2. - From each combination, Player 1's decision node is reached, with branches labeled 'Fold' and 'Bid'. - If Player 1 bids, Player 2's decision node follows, again with 'Fold' and 'Bid' branches. - At the end of the tree, terminal nodes represent the game's payoffs based on the decisions made and cards held.

Key concepts

Extensive-form game

In game theory, the extensive-form game is a way of representing games that highlights every possible sequence of moves. In our three-card poker game, this representation is crucial for illustrating each player's strategic options and the associated outcomes.

To craft an extensive-form game, start with a game tree. The root of this tree represents the initial state, typically starting with a chance move, or 'nature,' in the case where cards are dealt. This leads to various branches representing all feasible moves for the players. It's a visual and systematic way of capturing sequential choices within a game.

• Begin with the possible deals by nature, where each player receives one of the three cards. Every branch leading from the root represents a unique card combination.


• The next node shows Player 1's choice: to 'Fold' or to 'Bid'.


• If Player 1 chooses to bid, Player 2 then makes a similar decision: 'Fold' or 'Bid'.


• At terminal nodes, card reveal outcomes are displayed, completing the game.

In the extensive-form, attention to the structure and sequence of decisions is key. It captures every possible outcome in a structured framework, clearly illustrating the game's strategic trajectory.

Strategic decision making

Strategic decision-making involves evaluating potential actions by considering the possible responses of other players at each turn. In the three-card poker game, both Player 1 and Player 2 face critical decision points that shape the flow and outcome of the game.

Player 1 starts the sequence with a choice between folding and bidding. This decision is influenced by the card Player 1 has, and the anticipation of how Player 2 might respond. If Player 1 folds, the decision sequence ends immediately, leaving Player 2 as the winner of that round.

• Should Player 1 decide to bid, Player 2 then faces their strategic choice, again between folding or bidding.


• Players anticipate each other's likely actions, aligning their strategies to maximize their respective payoffs.


• It's essential for players to think several steps ahead, predicting the opponent's response in the extensive game structure.


• Through this anticipation, players make decisions that lead to achieving the most favorable outcomes within the game's rules.

Strategic decision-making is the heart of game theory, requiring players to weigh their choices against potential risks and benefits informed by their perceptions of other players' likely moves.

Payoff structure

The payoff structure in a game captures the rewards or penalties that players receive based on the decisions made and outcomes achieved. In our poker game, the payoff structure is a critical component that influences the players' strategies and ultimate choices. Each decision within the game's framework leads to specific tangible outcomes.

• If Player 1 folds immediately, the payoff is \((-1,1)\), meaning Player 1 loses their initial bet or ante, while Player 2 gains it.


• Conversely, if Player 1 bids and Player 2 folds, the result shifts to \((1,-1)\), benefiting Player 1 with a gain of the ante.


• When both players bid and reveal their cards, the player with the higher card takes the advantage with a \(2\) payoff, while the player with the lower card takes a \(-2\) hit. This implies a net loss for the weaker hand.

Understanding the payoff structure is essential for players. It drives strategic moves, guiding players to choose actions that maximize their expected gains or minimize potential losses. Payoff structure intricately weaves into decision-making, as players assess their probabilities and possible benefits at each decision node.