Thursday, December 15

Game Theory of The Price is Right: Part 2

In my last post, I discussed some game theory behind the Cliffhangers game on The Price is Right. And while watching the stupidity of Cliffhangers players got me angry, what really got me thinking was the bidding itself.

To get up on stage (which comes with a chance to win bigger prizes), four contestants take turns guessing the value of an item (usually several hundred dollars). The one with the closest guess to the item's actual value, without going over, wins the item and gets to go up on stage.

Frequently, people make really dumb guesses. Someone will bid, say, $420 for an item, and another contestant will subsequently bid $415, giving themselves a $5 window in which to win the item. But more interesting is whether or not contestants decide to bid $1 above someone else. Frequently, someone will bid, say, $475 after another contestant has bid $420. They could have bid $421 (and indeed this does happen, as in the video below), but in the unwritten etiquette of The Price is Right, it's viewed as a low blow.

Despite the evilness of the $1-top-up, the game theorist in me wondered why contestants — specifically, the fourth contestant — don't do it more often. I thought, perhaps, that it was because they are playing a repeated game (the three contestants who do not win get to bid again on a new item against a new fourth contestant, except after the sixth and final round of the show). If I'm playing against the same people again, then perhaps I don't want to be mean to them, since then they might be mean to me later.

But after further pondering, I realized this does not make sense. In the last round, repetition is not an issue, so the fourth contestant would want to bid $1 above someone else (or bid $1, if they think everyone else has overbid). Given that there is no incentive for contestants to cooperate in the last round, contestants shouldn't coopoerate in the fifth round (since there's no point to endearing themselves to their opponents for the sixth round). And if contestants shouldn't cooperate in the fifth round, there's no incentive to cooperate in the fourth round either, and so on. For game theorists, this is an example of a subgame perfect Nash equilibrium (solved using backwards induction).

So if fourth player's optimal bid is to do the evil $1-top-up, I found myself asking why this doesn't happen more often. Here are the possible reasons I though of:
  1. Some contestants are stupid. Economists don't like this answer, since it's easier to assume everyone is rational, but if you watch The Price is Right regularly, it's hard to dismiss this explanation.
  2. The show is televised; as much as contestants would love to win a karaoke set, contestants don't want to look like a huge jerk on national TV. This strikes me as a somewhat plausible explanation.
  3. Contestants don't like the item being presented (remember that if you lose, in most cases you'll be able to try again. So if I have no interest in winning the karaoke set, I may have an incentive to intentionally bid poorly and hope that the next item up for grabs is more appealing). This, however, is not a very convincing explanation, since contstants are not guaranteed another shot at winning, and the main attraction of winning is not getting the karaoke set, but getting the chance to win something bigger once the contestant gets up on stage.
  4. Contestants want to guess the exact price of the item. Contestants receive a bonus ($500, I think) for guessing the price of the item exactly. It may be that this incentive strongly influences bids. In the extreme case, if all I care about is the $500, it does not matter what my opponents bid, since whether I bid $419, $420 or $421, I would have the same odds, in theory, of guessing the exact price. But I'm not convinced by this explanation either, since the real prospect of winning bigger prizes should outweight the miniscule chance of obtaining the $500 incentive.
I wondered whether any game theorists had studied this before. And, low and behold, some have. Researchers Jonathan Berk, Eric Hughson and Kirk Vandezande did an eloquent study on Price is Right bidding (entitled The Price is Right, but are the Bids? An Investigation of Rational Decision Theory).

Reading the article, one can't help but gain an appreciation for the real beauty of the game theory behind The Price is Right. The math is somewhat complex, but it finishes in a clean result: in a world with identical contestants who act rationally, the fourth contestant wins one third (i.e. 3/9ths) of the time, and everyone else wins 2/9ths of the time.

In this perfect world, the first contestant bids highest, the second the next highest, the third the next highest, and the fourth bids $1. Players 1 through 3 evenly space their bids out across the probability distribution for the prize (think of this as the range of values that they think the price could be, taking into account how likely each value is to arise — for example, it's very likely that it's $500, somewhat possible it's $250 or $750, and very unlikely that it's $0 or $1,000). The study includes this graph, which may help visually-inclined readers:

My intuition was that, in an equilibrium, all players would evenly space out their bids, so I was confused by the theory at first. But the reason contestants don't evenly space out their bid is that the last player gets the trump card: he or she could bid $1 more than someone else, without having to worry about someone else doing the same to them (this round, at least). So if the first three contestants bid at 0.75, 0.5, and 0.25 in the graph above, then the fourth contestant could bid 0, 0.2500001, 0.5000001 or 0.7500001, and have for all intents and purposes an equal chance of winning that sequence of bidding.

But bidding 0 isn't the best strategy, since if the fourth contestant bids $1 more than one of the previous contestants, he or she has the same chance of winning the round, but there is also the chance that everyone overbids, in which case the players have to bid again and get a shot at winning. In other words, the fourth bidder is not indifferent between bidding a dollar and bidding a dollar more than someone else when bids are evenly spaced, since if they bid a dollar more than someone else, they get two chances at winning (once the first time, and again if everyone overbids). They only get one shot at winning when they bid $1.

Thus, the other players have to concede a little bit to the fourth contestant in order to coax him or her into bidding $1, thereby preventing the fourth contestant from doing an evil top-up bid.

But does the theory play out in real life? Not at all. The study examined dozens of Price is Right episodes and basically concluded that the average contestant is stupid:
"Our results indicate that rational decision theory cannot explain contestant behavior on The Price Is Right. Even when faced with relatively simple problems, we demonstrate that some (indeed most) contestants do not deduce the optimal strategy."
A follow-up study by Paul Healy and Charles Noussair found similar results, namely that the Price is Right is to complicated for people to figure out (although if you cut it down to three contestants, remove the possibility of rebidding if everyone overbids, and let people play a whole bunch of times, they start to get the hang of it).

So the venerable Happy Gilmore might have summed it up best: when it comes to contestants' use of game theory on the Price is Right, more often than not, "The price is wrong, b@#$%."