I'm trying to find the Nash Equilibrium of a simple betting game, and have come up with a very surprising result which I'd like to solicit comment on.

The game is simple: Two players each receive a secret real number selected randomly (uniform distribution) from $(0,1)$. Each player secretly chooses to *bet* or *fold*. If either player folds, each player receives $0$. If both players bet, the player with the higher number receives $1 + \alpha$, and the player with the lower number recieves $-1$ (i.e. looses 1).

(Note: To simplify discussion, I am ignoring ties and other probability 0 events.)

If there is no difference in expected values, both players would slightly prefer to fold than to bet.

My result: No matter what $\alpha$ is, the only Nash Equilibrium is for both players to *always fold*!

Proof sketch:

- Assume there exists a value of $\tau$ that Player 1 will bet iff he believes his prob. of winning is above $\tau$. (Generally, $\tau * (1 + \alpha) > (1 - \tau)$.)
- Before the game-theoretic regress, Player 1 computes that he will have prob. $> \tau$ if his secret number is $> k$.
- By symmetry, Player 2 will only bet if his value if also $> \tau$, which means that his secret number is $>k$.
- Which of courses raises Player 1's value of $k$ in an infinite regress. The formula is $k_{n+1} = (1-k_n)*\tau + k_n$, which converges at $1$.

Is this correct? It means that even if $\alpha$ is very large, both players will always fold, which is highly counter-intuitive.

*NOTE:* In researching this, I came across this excellent post Finding the Nash Equilibrium of $0-1$ poker with one betting round and https://groups.google.com/forum/?fromgroups#!topic/rec.gambling.poker/wFZ_aJqVY_A%5B1-25%5D , which address similar problems, though neither seems to address this simple game with such a surprising result.

**UPDATE: What's unique and still unexplained?**

The responses have confirmed my result. However, I believe **the question still stands**. I believe the result is unique in that it shows a huge discrepancy between game theory and real life rational behavior, which none of the responses addresed or even explored. Let me illustrate:

You and an opponent are each dealt a hand and shown a pot of $100. After receiving your hand, you can choose to play, which costs one dollar, or walk away. Afterwards, your opponent is given the same choice. If both bet, winner takes all. If either one walks away, the pot goes back to the house.

You see that you have a very good hand (4 of a kind, let's say). I would certainly play. I do not know a single person that I think would refuse to play. I would play even if my opponent was a Fields medal laureate. If someone told me they'd refuse, I would think it foolish. Yet, the only Nash equilibrium behavior (that is, the only rational behavior given common knowledge of both players rationality) is to walk away. This is true even if the pot was $100,000.

**This gross discrepancy between what even seemingly highly rational people would do in real life and what game theory tells us is extraordinary.**

This is not true for the prisoners dilemma (cited by some responses), where many people would indeed confess in real life. And it has nothing to do with the fact that the game doesn't model real life poker when folding (as other responses mentioned). **It's a case where game theory tells us one thing, and even the most rational intelligent players do something else.**

**To answer this question, I'd like to see this discrepancy explained or even explored**. Or even similar examples which are discussed elsewhere cited.

traveller's dilemmaat faculty.arts.ubc.ca/rjohns/basu.pdf. – Michael Greinecker Aug 20 '12 at 19:10