Question

5\. Consider a game inspired by the long-running television program The Price Is Right. Three players have to guess the price \(x\) of an object. It is common knowledge that \(x\) is a random variable that is distributed uniformly over the integers \(1,2, \ldots, 9\). That is, with probability \(1 / 9\) the value of \(x\) is 1 , with probability \(1 / 9\) the value of \(x\) is 2 , and so on. Thus, each player guesses a number from the set \(\\{1,2,3,4,5,6,7,8,9\\}\). The game runs as follows: First, player 1 chooses a number \(n_{1} \in\\{1,2\), \(3,4,5,6,7,8,9\\}\), which the other players observe. Then player 2 chooses a number \(n_{2}\) that is observed by the other players. Player 2 is not allowed to select the number already chosen by player 1 ; that is, \(n_{2}\) must be different from \(n_{1}\). Player 3 then selects a number \(n_{3}\), where \(n_{3}\) must be different from \(n_{1}\) and \(n_{2}\). Finally, the number \(x\) is drawn and the player whose guess is closest to \(x\) without going over wins \(\$ 1000\); the other players get 0 . That is, the winner must have guessed \(x\) correctly or have guessed a lower number that is closest to \(x\). If all of the players' guesses are above \(x\), then everyone gets \(0 .\) (a) Suppose you are player 2 and you know that player 3 is sequentially rational. If player 1 selected \(n_{1}=1\), what number should you pick to maximize your expected payoff? (b) Again suppose you are player 2 and you know that player 3 is sequentially rational. If player 1 selected \(n_{1}=5\), what number should you pick to maximize your expected payoff? (c) Suppose you are player 1 and you know that the other two players are sequentially rational. What number should you pick to maximize your expected payoff?

Short answer

(a) Pick 2; (b) Pick 6; (c) Pick 5.

Step-by-step solution

Understanding the Uniform Distribution

The object price \(x\) is uniformly distributed over the integers 1 to 9. This means each integer has an equal probability of \(\frac{1}{9}\). Therefore, there is no prior information that favors any number over another based on price alone.

Player 2's Choice if Player 1 picks 1

If Player 1 chooses \(n_1 = 1\), Player 2 should pick the next lowest available number close to the middle of the range to maximize their chances for numbers \(x\) above 1. The rational choice is \(n_2 = 2\), as it allows Player 2 to be closest to the lower numbers without giving too much advantage to Player 3 who will have more flexibility in spreading their choice.

Player 2's Choice if Player 1 picks 5

If Player 1 selects \(n_1 = 5\), Player 2 should consider numbers higher than 5 to position themselves on the upper half of the possible values, allowing Player 3 to cover possibilities below 5. Thus, a sequentially rational player would choose \(n_2 = 6\). This maximizes Player 2's expected payoff for values \(x\geq 6\).

Player 1's optimal choice knowing others are rational

Player 1 needs to select a number that gives them a balanced position considering that Players 2 and 3 will act rationally. When choosing \(n_1 = 5\), Player 1 positions themselves in the middle of the range, increasing the chance to win if \(x\) is close to the middle of the distribution, which maximizes Player 1’s potential to win given the uniform distribution.

Key concepts

Sequential Rationality

In game theory, sequential rationality refers to the concept where each player's decision takes into account the future actions of others. It emphasizes that players anticipate how their actions influence subsequent choices made by competitors.

Applied to the problem at hand, if Player 1 picks a number, Player 2 must select their number understanding that Player 3 will act rationally afterward. They must predict the counteractions of other players to optimize their own outcomes. Consistently, Player 2, for example, should choose numbers that best position them against future player guesses, knowing what Player 1 selected and foreseeing Player 3’s rational moves.

In essence, this means that players do not act in isolation, but rather think ahead and evaluate how others will respond to their strategic choices.

Uniform Distribution

Uniform distribution implies that every outcome in a given range has an equal probability of occurring. In this game, the object price, represented as \( x \), is uniformly distributed over the integers from 1 to 9.

For each game play, every number has a probability of \( \frac{1}{9} \) to be the correct one. This essential characteristic of the game keeps all numbers equally fair and unpredictable by price alone. Players cannot infer which number is more likely to appear based on distribution, thus, they rely on strategic choices rather than probabilistic prediction based on distribution favorability.

This ensures that every player's strategic decision making is on equal footing in terms of probability, shifting the game balance towards sequential reasoning and adaptive immersion.

Expected Payoff

Expected payoff refers to the average outcome a player anticipates considering the various probabilities of different results. It’s a critical element when deciding how to choose numbers in this game.

For instance, Player 2 analyzes their expected payoff by weighing probabilities of outcomes based on Player 1’s choice. If Player 1 is at \( n_1 = 1 \), and Player 2 chooses \( n_2 = 2 \), their expected payoff increases by covering numbers close to the start of the sequence. They get positioned to win by having guessed directly or being the closest below the drawn \( x \).

Expected payoff calculation is part probability, part strategy choice. It influences the rational decision-making process as players calculate risks and positions to statistically optimize their chances of winning the prize.

Strategic Decision Making

Strategic decision-making in game theory involves choosing actions that maximize an individual's payoff while considering possible responses from opponents. It requires players to analyze their options and plan for future outcomes.

In the given game, strategic decision making is seen when Player 2 chooses their number based on Player 1's selection and anticipates Player 3’s potential to pick the remaining options. Picking proactively while contemplating others' choices is key.

This involves a keen understanding of not only maximizing individual gain but also minimizing chances for others to surpass. Players must weigh different scenarios: what happens if their chosen number does not win? How can they prevent opponents from being in the best position? Ultimately, the essence of strategic decision making is in the intelligent and predictive adaptation to dynamic game conditions, leading players closer to optimal outcomes.