# Strategies for Liar's Poker

This question is only tangentially related to quantitative finance. Scott Patterson's book The Quants describes how a quant at Kidder Peabody figured out a strategy to playing Liar's Poker in the late 80s. This strategy spread among quants at investment banks and led to it not being played any more.

Here is a simple description of the basics of the game for those not familiar: http://www.investopedia.com/terms/l/liars-poker.asp#axzz27gZIYnwx

The strategy described in Patterson's book was more or less to use the information from your own bill to have more confidence in making large bets. For instance, if you have two 3s on your bill when the previous bid was four 9s, then rather than bid five 3s you should bid 10 or more 3s (when there were 10 players). How much you should increase your bet in the strategy was likely based on Bayesian reasoning, though the book does not go into that much detail.

Is this really the best strategy or only the best strategy under some conditions?

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Short answer: there are multiple optimal mixed (i.e. non-deterministic) strategies.

Long answer: There is an equivalent game, called bluff or liar's dice, which is played using dice. Each player has a number of dice, and can see only their own rolls. The game consists of claims on the whole pool of the dice of all players, for example "there are at least 4 dice showing 5". The players take turns where they can either call or raise. A raise means to either increase the number each dice has to show, or increase the number of dice.

For two players and since liar's dice is a zero-sum game, you can apply the mini-max argument to show that it must have at least one optimal strategy. For more players Nash showed that there is always a set of optimal strategies at Nash equilibrium. These can't be deterministic strategies though, since a deterministic strategy would reveal information about the players dice to the opponent. Lanctot and Long use a computer program to find a solution for 2 the dice, 2 people game of bluff. They also show how Liar's dice can be expressed as a linear program. Since linar programs have have convex feasible regions, Liar's dice has multiple pairs of optimal strategies at Nash equilibrium.

We don't know much about how these optimal strategies look like, though.

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The strategy was discussed in the book "The Poker Face of Wall Street" by Aaron Brown (p258-267). Let's say that \$20 bills are put into a hat and then drawn randomly. The serial number on the bill becomes your "hand." According to Brown the key to the game was the position of the betters. The hierarchy determined the betting order. So, the senior people bet first and junior traders always bet last. This was a disadvantage for the juniors if they played a naive strategy. What Brown discovered was a system that didn't require cooperation between people but escalated the bids very quickly thus jamming the early bidders when the betting came back around to them.

As an aside, I liked Brown's book. It's a quick, fun read. He seems like an interesting guy who'd kill me at poker.

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