Question

In the envelope game, there are two players and two envelopes. One of the envelopes is marked "player 1 ", and the other is marked "player 2 ." At the beginning of the game, each envelope contains one dollar. Player 1 is given the choice between stopping the game and continuing. If he chooses to stop, then each player receives the money in his own envelope and the game ends. If player 1 chooses to continue, then a dollar is removed from his envelope and two dollars are added to player 2's envelope. Then player 2 must choose between stopping the game and continuing. If he stops, then the game ends and each player keeps the money in his own envelope. If player 2 elects to continue, then a dollar is removed from his envelope and two dollars are added to player l's envelope. Play continues like this, alternating between the players, until either one of them decides to stop or $k$ rounds of play have elapsed. If neither player chooses to stop by the end of the $k$ th round, then both players obtain zero. Assume players want to maximize the amount of money they earn. (a) Draw this game's extensive-form tree for $k=5$. (b) Use backward induction to find the subgame perfect equilibrium. (c) Describe the backward induction outcome of this game for any finite integer $k$.

Short answer

Player 1 should stop at or before decision points where payoff exceeds future risks. For any k, stop early to avoid ending at zero.

Step-by-step solution

Understanding the Initial Game Setup

Initially, Player 1 and Player 2 both have $1 each. They have to decide at each turn whether to stop or continue playing. Continuing allows transfer between players but the final goal is to maximize their own money.

Constructing the Extensive-Form Tree for k = 5

Draw a tree starting with Player 1's decision node at the top level. If Player 1 continues, a dollar is deducted from their amount and two dollars are added to Player 2's envelope. This transfer is alternated up to five rounds. For each level of decision (from Player 1 to Player 2 and vice versa), two branches split: one for 'stop' and one for 'continue'. If five rounds are completed without stopping, both receive $0.

Implement Backward Induction

Backward induction involves analyzing the game from the last possible action back to the first. If reached round 5 and no player stops, players receive $0. For round 5 decisions: stopping gives (current balance), continuing gives a risk/reward of cycling back to them with potential zeros. Work backwards, deciding based on maximizing expected payout.

Solving Step-by-Step for k = 5

Identify optimal stopping points through backward induction. E.g., at round 4 if payoff is greater by stopping than at round 5 potentially getting zero, prefer stopping. Extrapolating this, players tend to prefer stopping at earlier reachable points when payoff exceeds risks involved or zero payoffs.

Generalizing the Outcome for Any k

For any finite integer, players will prioritize stopping at a decision that yields more than playing through all rounds risking hitting the zero payoff. As k increases, the optimal play is determined increasingly earlier as zero payoff at k-th round looms larger.

Key concepts

Backward Induction

Backward induction is a method used in game theory to determine the optimal strategy for players making sequential decisions. This technique involves analyzing the game from the end, or last possible decision point, moving backwards to the beginning.

By evaluating the payoffs from the final possible actions first, players deduce the logical moves that lead to their best outcomes. In theory, this approach ensures that players end up making decisions that yield the greatest benefit when considering all potential future actions.

• Players analyze the end game first.
• Decisions are made to maximize the final payoff.
• It's a rational way to solve games with a clear endpoint.

This approach is particularly beneficial in extensive-form games with finite moves and clear outcomes.

Extensive-Form Game

An extensive-form game represents a game in a tree structure, showing how decisions are sequentially made by players. Each node signifies a decision point, with branches indicating possible actions and payoffs.

This structure helps players visualize the sequence of actions and possible outcomes. It is essential when dealing with complex strategic decisions as it presents an organized way to map out every possibility and corresponding result in the game.

• Displays decisions and their sequence.
• Shows possible moves by players at each node.
• Includes payoffs associated with different paths.

By viewing a game in this form, players and analysts can better comprehend the potential strategic moves involved.

Subgame Perfect Equilibrium

In game theory, a subgame perfect equilibrium is an optimal strategy that reflects the best possible outcome for each player at every stage of the game. This concept excludes any non-credible threats or unlikely strategies players might make.

It involves players executing optimal actions at each subgame or decision node based on their rational expectations, essentially reinforcing sound decision-making from start to finish. This kind of equilibrium ensures no player benefits from deviating at any stage if others do not.

• Optimal strategy at all decision stages.
• Eliminates unlikely or irrational strategies.
• Ensures consistency and rationality in decision-making.

Finding a subgame perfect equilibrium can involve using backward induction, as players attempt to foresee the best strategies throughout the game's entire sequence.

Strategic Decision Making

Strategic decision making in game theory involves players weighing their choices to achieve the best possible monetary outcome while considering the potential moves of other players.

Players must anticipate the reactions of others, akin to a game of chess, strategizing several moves ahead. This requires understanding all aspects of the decision-making landscape, including potential payoffs, risks, and others' objectives.

• Focuses on maximizing individual outcomes.
• Considers opponents' potential strategies.
• Involves risk assessment and probability of different scenarios.

Efficient strategic decision-making ensures that players are better equipped to make informed choices that could alter the course of the game favorably.