Las Vegas and game theory: an application of the Prisoner’s Dilemma

The Prisoner’s Dilemma is a great example from game theory. The game illustrates why individuals might not cooperate even if it is their best interest to do so.

I will briefly summarize the game below, but if you’re familiar with the Prisoner’s dilemma you can safely skip ahead to the section below about poker tournaments.

The game

In the classic story, two suspects are apprehended. The police have insufficient evidence and need at least one confession to convict. The police come up with the following questioning scheme.

First, they interrogate the suspects separately to prevent communication. Second, they offer a conditional sentence to each suspect:

–If a suspect testifies, and the partner remains silent, then the testifier goes free and the other gets a 10-year prison sentence
–If both confess, then each gets a reduced 5-year sentence
–If both stay silent, then they can only be charged with a minor offense and will face a 6-month sentence

What should the prisoners’ do? It is tempting for both to stay silent and receive the minimum. But each is worried the partner might squeal and walk away, leaving the other to serve a 10-year sentence.

The strategic method is to examine each choice conditional on the other’s decision. And here is what the logic shows:

–If the partner stays silent, then it would be best to confess (no sentence) rather than stay silent (6 months)

–If the partner confesses, then it would be best to confess as well (5 years) rather than stay silent (10 years)

And this is the devious trap set up by the police. It is readily apparent that each suspect does better by confessing, and so both will confess and accept 5-year sentences.

The outcome is so devastating it feels like the police have cheated. It just feels bad that the suspects could do much better, if only they had a little more trust. But alas that is the trap the police knowingly have set.

The police win because they divided their opponents and created seemingly fair but ultimately devious conditions. And this is a lesson Las Vegas casinos have taken to heart.

Poker tournaments

My friend Jamie is a professional poker player, and he came across a great example along the lines of the Prisoner’s Dilemma.

Here is what he reports:

I played a poker tournament at Caesar’s Palace last night with the
following setup: The buy-in is $65, which gets you 2500 chips. There
is also the option to buy an additional 500 chips for $5 more, giving
you a total of 3000 chips for $70. At 1 cent/chip, this add-on sounds
like a great bargain compared to the 2.6 cents/chip of the regular
buy-in.

The kicker is that the house keeps the entire $5 add-on fee; none of
it goes into the prize pool.

It is the last sentence that reveals the devious set-up. The casinos are playing on individual incentives just like the police did to the suspects.

Since none of the extra money goes to the pot, the ideal solution would be for no one to buy the extra chips. The pot would stay the same and everyone’s chip stack would be equal at the start. This is analogous to both suspects staying silent in the Prisoner’s Dilemma.

And that is the rub. No one can be sure the other players will cooperate. It is therefore necessary analyze the decision based on what other’s might do:

–If the other players do the standard buy-in, then it would be best to buy more chips (bigger stack = more power) rather than to do the standard buy-in (equal chips)

–If the other players do the extra buy-in, then it would be best to do the extra buy-in as well (equal chips) rather than to do the standard buy-in (small chip stack)

This is a devious trick to get everyone to contribute extra money to the house.

Or, as Jamie puts it:

So while each entrant is motivated to purchase the additional chips
at a big discount, the net effect is that everyone still starts with
the same size stack, and the house is $5 richer for each entry.
Though the players are best off if everyone refuses the add-on,
any one player can gain an advantage by buying it, and thus all do.

This is a devious yet ingenious application of game theory, and I hear
all the major Vegas casinos have a similar policy.

This is a remarkable example as it about as close to textbook game theory as I’ve seen. It is apparent Jamie bought the extra chips, and I have no doubt I would do the same.

Anyone else see this practice in poker tournaments? Any ideas on how players could retaliate?

Discussion questions

1. How would you apply this lesson to selling raffle or lottery tickets? an office pool?

2. What could the players do to change the game?

3. How might the game change if the tournament is repeated annually?

4. What would happen if the extra chips could be resold among players?