### I'd Like to Bid $1, Bob!

by Ben Resnick
In a previous post, we used The Price Is Right as a starting point for a discussion of probability theory and decision-making analysis. But the opportunities to learn from the show hardly stop there. Another practical game theory application can be observed six times a show – when contestants “bid” in an effort to win a prize and to get on stage from “contestants’ row.”

Each time the game is played, four contestants are shown a prize without being told its price. Each contestant takes their turn announcing a single bid, or guess, as to the value of the prize (rounded to the nearest dollar). All bids are known to all the other players as soon as they are made, and no player may bid the same value as another. After all four bids have been made, the player whose bid is closest to the actual retail price of the prize without going over wins both the prize and the opportunity to come on stage and play a “pricing game” in order to win more valuable prizes. If all four contestants guess a price higher than the price of the prize, all bids are erased and the game is played again. In the case that one of the four players actually guesses the exact value of the prize, they receive an additional cash bonus. For the remaining purposes of this analysis though, we will ignore the cash bonus and simply focus the analysis on a strategy to maximize the chances of winning the prize and going on stage. The game is most easily modeled from the perspective of the fourth player, so we will base our analysis on his perspective.

Suppose the first three players bid b1, b2, and b3; assume these are listed in increasing order because only the value of each of the first three bids matters to the fourth player, not which bid was made by which player. The fourth player will then try to pick a value, b4, that is priced closer to the true price of the prize (call it P) than b1, b2, and b3, without going over. Assuming none of the other three players has bid the exact price, P will fall in one of the following four intervals: (1,b1-1), (b1+1,b2-1), (b2+1,b3-1), (b3+1, ∞).

The optimal strategy for the fourth player is to pick what range they think contains P, then bid the lowest value in that range. It is important to note that any bid higher than this but within the same range does not help him, but it does increase the chance that he goes “over” and loses automatically. Suppose, for example, that the fourth player believes that P= $1,000 and b3=$900 (with b1 and b2 defined to be less than $900). The fourth player should bid exactly $901. With a belief that the price is around $1,000, any bid lower than $900 gives the player who guessed b3 the best chance to win, and any bid higher than $901 gives the same player more chances to win if it turns out the fourth player’s belief about the price is an overestimate. In order to act optimally, the fourth player should always bid either: $1, b1+1, b2+1, or, b3+1. Any other bid makes the fourth player strictly worse off; taking away values for P that make the fourth player win, without providing other values that make him win.

Discussion Questions

1. Given the fourth player’s optimal strategy, how should the third player pick their bid? Keep in mind, when making his choice, the third player knows b1 and b2, as well as the fourth player’s strategy. What method can you use to solve for each player’s optimal strategy in this game?

2. What would you expect to happen if all players wrote down their bids simultaneously and did not know the other players’ guesses when making their own selection?

3. Suppose the cash bonus for bidding the exact value of the prize were very large, so large that players would be willing to risk losing the game for a chance to collect the bonus. How would this change the optimal bidding strategy? How would each player’s risk aversion factor into his decision?

4. Many times on the show, we do not observe this optimal bidding strategy by the fourth player. One possible explanation is that players do not want to appear cutthroat and greedy by bidding a single dollar more than an opponent (thus giving the opponent only one way to win: if his bid is exactly the true price). How would the optimal strategy change if you add reputation costs?

Each time the game is played, four contestants are shown a prize without being told its price. Each contestant takes their turn announcing a single bid, or guess, as to the value of the prize (rounded to the nearest dollar). All bids are known to all the other players as soon as they are made, and no player may bid the same value as another. After all four bids have been made, the player whose bid is closest to the actual retail price of the prize without going over wins both the prize and the opportunity to come on stage and play a “pricing game” in order to win more valuable prizes. If all four contestants guess a price higher than the price of the prize, all bids are erased and the game is played again. In the case that one of the four players actually guesses the exact value of the prize, they receive an additional cash bonus. For the remaining purposes of this analysis though, we will ignore the cash bonus and simply focus the analysis on a strategy to maximize the chances of winning the prize and going on stage. The game is most easily modeled from the perspective of the fourth player, so we will base our analysis on his perspective.

Suppose the first three players bid b1, b2, and b3; assume these are listed in increasing order because only the value of each of the first three bids matters to the fourth player, not which bid was made by which player. The fourth player will then try to pick a value, b4, that is priced closer to the true price of the prize (call it P) than b1, b2, and b3, without going over. Assuming none of the other three players has bid the exact price, P will fall in one of the following four intervals: (1,b1-1), (b1+1,b2-1), (b2+1,b3-1), (b3+1, ∞).

The optimal strategy for the fourth player is to pick what range they think contains P, then bid the lowest value in that range. It is important to note that any bid higher than this but within the same range does not help him, but it does increase the chance that he goes “over” and loses automatically. Suppose, for example, that the fourth player believes that P= $1,000 and b3=$900 (with b1 and b2 defined to be less than $900). The fourth player should bid exactly $901. With a belief that the price is around $1,000, any bid lower than $900 gives the player who guessed b3 the best chance to win, and any bid higher than $901 gives the same player more chances to win if it turns out the fourth player’s belief about the price is an overestimate. In order to act optimally, the fourth player should always bid either: $1, b1+1, b2+1, or, b3+1. Any other bid makes the fourth player strictly worse off; taking away values for P that make the fourth player win, without providing other values that make him win.

Discussion Questions

1. Given the fourth player’s optimal strategy, how should the third player pick their bid? Keep in mind, when making his choice, the third player knows b1 and b2, as well as the fourth player’s strategy. What method can you use to solve for each player’s optimal strategy in this game?

2. What would you expect to happen if all players wrote down their bids simultaneously and did not know the other players’ guesses when making their own selection?

3. Suppose the cash bonus for bidding the exact value of the prize were very large, so large that players would be willing to risk losing the game for a chance to collect the bonus. How would this change the optimal bidding strategy? How would each player’s risk aversion factor into his decision?

4. Many times on the show, we do not observe this optimal bidding strategy by the fourth player. One possible explanation is that players do not want to appear cutthroat and greedy by bidding a single dollar more than an opponent (thus giving the opponent only one way to win: if his bid is exactly the true price). How would the optimal strategy change if you add reputation costs?

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