John Fremlin's blog: The Right Price for the Price is Right: Optimal Bidding

The famous TV game show, The Price is Right had an excellent one-bid game where four players would take turns to give a guess for the price of an item (no guesses can be repeated). Any guess higher than the item price (an overbid) was discarded and then the closest remaining lower guess would determine the winner. Sometimes there was an exact prize bonus too.

This game has been studied extensively. When I first heard the rules, I wrongly intuited that the overbid condition would make it easier for the first player: quite the opposite.

The first simplification to make thinking about the problem easier is to reduce it to assume that all participants share the same model of the distribution of the price. Provided they are not sure about the price, then this resolves to guessing the closest value under a fair many sided die or roulette wheel spin.

Suppose we first consider the case where if all contestants overbid, then nobody wins the prize and the game is not repeated. The strategy of the last player is very easy: just pick the one more than a previous bid, or pick the lowest possible bid, whichever one has the highest probability weight between it and the next highest bid or infinity if it would be the highest bid.

Inducing backwards, the second from last player must consider that the last player will take almost all the probability mass from his bid, by bidding just above it, if that bid has the largest probability region above it. Therefore he or she must choose a bid that leaves another more tempting region for the last player to take. This implies that each preceding player must take at most their fair share - that is the first player must bid at min{b : P(X ≥ b) ≤ 1/n}, or for four players on a roulette wheel from 1-36, he or she would bet just exactly than 1/4 from the end, or 28 (winning on the 9 numbers, 28-36).

Suppose he or she bid anything less - even just one less, 27. Then he or she would have 10 winning numbers, but the next player might be tempted to take the range 28-36 which has 9 winning numbers and leaves the big region 1-26 to be safely split between the third and fourth.

Otherwise bidding below, the lowest the second player could safely go is 18, and also only get nine numbers. As there is no advantage to this, he or she might as well go for 28.

We therefore assume the first player picks 28, so the second player will take 19-27 (nine numbers), leaving the third player to take 10-18, nine numbers.

Consequently everybody will get an equal share. This analysis differs extremely from The Price Is Right, But Are the Bids? An Investigation of Rational Decision Theory by Berk, Hughson, and Vandervonde. In their version they fail to consider the discrete nature of the problem and assume that in cutting off another player (guessing just above another player) one receives the entire probability mass of that player. However, of course, one cannot repeat the exact guess so there is a little that previous player gets to keep.

As another difference, they consider a game where in the case where everybody overbids, the bidding is restarted (with an implicit maximum of the previous lowest bid, and in the same player order). In this version, the last player has an incentive to not bid 1, as the game will then restart. Then for all possible prices outcomes below the lowest bid, the last player will receive a probability mass P_lastwin that he or she would win after a restart. This gives a probability mass bonus of P_lastwinP(min(b_i) > X) to the interval obtained by cutting off another player. Assuming as this paper does, that the space of bids is continuous and not discrete this means that the last player would cut off the best previous player's interval (so in this continuous bid version where each bid has measure zero, taking the entire probability mass) getting at least 1/3 of the mass if not overbid, and otherwise gambling on a restart. This implies that P_lastwin is 1/3.

In practice, contestants are very nervous about overbidding and in 54% of cases the winning last bet is to cut off the highest bid.