Question

Represent the following game in the extensive form. There are three players, numbered 1,2 , and 3 . At the beginning of the game, players 1 and 2 simultaneously make decisions, each choosing between " \(X\) " and " \(Y\)." If they both choose " \(\mathrm{X}\)," then the game ends and the payoff vector is \((1,0,0)\); that is, player 1 gets 1 , player 2 gets 0 , and player 3 gets 0 . If they both choose " \(Y\)," then the game ends and the payoff vector is \((0,1,0)\); that is, player 2 gets 1 and the other players get 0 . If one player chooses "X" while the other chooses "Y," then player 3 must guess which of the players selected " \(\mathrm{X}\) "; that is, player 3 must choose between " 1 " and " \(2 .\) " Player 3 makes his selection knowing only that the game did not end after the choices of players 1 and 2 . If player 3 guesses correctly, then he and the player who selected " \(\mathrm{X}\) " each obtains a payoff of 2 , and the player who selected " \(Y\) " gets 0 . If player 3 guesses incorrectly, then everyone gets a payoff of 0 .

Short answer

The game involves sequential decisions with initial simultaneous moves by players 1 and 2, followed by player 3's decision to guess when needed. Payoffs depend on correct guesses. A tree diagram visually represents this game.

Step-by-step solution

Determine Initial Actions

The game begins with players 1 and 2 making simultaneous choices. They each have two options: "X" or "Y". So, there are four possible combinations of their actions: (X,X), (X,Y), (Y,X), and (Y,Y).

Assign Payoffs for Simultaneous Choices

For the combination of actions: - (X, X): The game ends with payoffs (1, 0, 0) as player 1 receives 1 and players 2 and 3 receive 0. - (Y, Y): The game ends with payoffs (0, 1, 0), with player 2 receiving 1 and others receiving 0.

Determine Player 3's Decision

For the mixed choices (X, Y) and (Y, X), the game continues. Player 3 must then decide which among the two players selected X. - If the actions are (X, Y), player 3 chooses between 1 and 2, meaning they have to guess that player 1 selected X. - If the actions are (Y, X), player 3 chooses between 1 and 2, assuming player 2 selected X.

Outcome Based on Player 3's Guess

If player 3 guesses correctly: - In case (X, Y) and guesses "1": Payoffs are (2, 0, 2). - In case (Y, X) and guesses "2": Payoffs are (0, 2, 2). If player 3 guesses incorrectly, there is no benefit for anyone: - In cases (X, Y) guesses "2" or (Y, X) guesses "1": Payoffs are (0, 0, 0).

Visual Representation

Translate the decisions into a tree where: 1. The first layer involves nodes branch for the decisions by players 1 and 2. 2. End decisions for (X,X) and (Y,Y) as terminal nodes. 3. For (X,Y) and (Y,X), branches continue to player 3's guessing node. 4. Depending on player 3's choice, the tree will show terminal outcomes with corresponding payoffs.

Key concepts

Game Theory

Game theory is a mathematical framework used to analyze decision-making among different entities or 'players'. It helps understand strategic interactions where the outcome for each participant depends on the actions of all involved. In our game, we have three players, each making independent choices that collectively determine everyone’s payoff. This game's structure is an example of an "extensive form game," meaning it can be represented as a decision tree that visually maps out the actions and possible outcomes at different stages. Understanding the strategic interplay in game theory involves recognizing each player's available options and predicting potential responses from others. This method is crucial for predicting and explaining behavior in economic, political, and social environments.
In this context, game players need to foresee the reactions and counter-reactions of others to optimize their strategies and maximize payoffs.

Strategic Decision Making

Strategic decision making in extensive form games involves players making choices at different points in time, often with some level of interdependence. Each player's choice can significantly impact the others, necessitating foresight and strategy. In our example, players 1 and 2 make simultaneous decisions, creating initial conditions for the game’s outcome. The subsequent decision by player 3 is contingent upon the earlier choices of players 1 and 2. This structure requires player 3 to use available information about the results of players 1 and 2’s actions to make an informed guess. Such scenarios showcase the essence of strategic decision making, where understanding not only your own options but also the likely choices of others can drastically alter the final payoffs.
This intertwined decision process highlights the need for each player to think several steps ahead, anticipating possible strategies the opponents might adopt.

Payoff Matrix

The payoff matrix is a key tool in game theory used to summarize the potential outcomes of a game. It illustrates the payoffs each player gets from a combination of different strategy choices. In simple terms, it is a box or grid that displays the gains or losses across all scenarios based on players' actions. In the given exercise, when both players 1 and 2 choose "X", the payoff is (1, 0, 0), meaning player 1 gains while the others get nothing. Conversely, both choosing "Y" results in a new payoff: (0, 1, 0), benefiting player 2 alone. Mixed choices lead to additional complexity where player 3's guess determines the final payoffs. - For an accurate guess in (X, Y) by player 3, the payoff is (2, 0, 2).
- In (Y, X), a correct guess gives (0, 2, 2).
- If player 3 guesses wrongly, everyone ends with (0, 0, 0). Understanding this payoff structure allows players to assess the risks and rewards of different strategies, aiding in optimal decision making.

Player Strategies

Player strategies in game theory refer to the specific plans or actions chosen to achieve desired outcomes. They can be simple, involving a single move, or complex, involving a series of contingent actions. In this game, players 1 and 2 start with straightforward strategies: choose between 'X' or 'Y', which sets the foundation for the game's progression. Their decisions directly affect player 3's strategy, who must decide based on the revealed information post their simultaneous choices. Each strategy comes with inherent risks and potential rewards, which are influenced by the guesses of player 3 in mixed scenarios. By understanding potential strategies and their associated payoffs, players can better navigate the complex interactions within the game to optimize outcomes. - If players 1 and 2 choose strategically, expecting player 3's reactions, they can potentially leverage this dynamic to their benefit.
- Player 3’s strategy depends on probability assessments and reacting to predecessors' choices. This dynamic interplay emphasizes how strategies are formulated, dependent on anticipated moves and guesses from other players.