ON THE BOREL AND VON
NEUMANN POKER MODELS
Comparison with Real Poker

Real Poker:
 Around
2.6 million possible hands for 5 card stud
 Hands somewhat independent for Texas Hold ‘em


Let’s assume probability of hands comes from a
uniform distribution in [0,1]
Assume probabilities are independent
The Poker Models

La Relance Rules:
 Each
player puts in 1 ante before seeing his number
 Each player then sees his/her number
 Player 1 chooses to bet B/fold
 Player 2 chooses to call/fold
 Whoever has the largest number wins.

von Neumann Rules:
 Player
1 chooses to bet B/check immediately
 Everything else same as La Relance
The Poker Models

http://www.cs.virginia.edu/~mky7b/cs6501poker/
rng.html
La Relance


Who has the edge, P1 or P2? Why?
Betting tree:
La Relance

The optimal strategy and value of the game:
 Consider
the optimal strategy for player 2 first. It’s no
reason for player 2 to bluff/slow roll.
 Assume
the optimal strategy for player 2 is:
 Bet
when Y>c
 Fold when Y<c
 Nash’s
Equilibrium
La Relance


P2 should choose appropriate c so that P1’s decision
does not affect P2:
If PI has some hand X<c, the decision he makes should
not affect the game’s outcome.

Suppose PI bets B
P1 wins 1 if P2 has Y<c (since he folds ‘optimally’)
 P1 loses B+1 if P2 has Y>c (since he calls ‘optimally’)


Suppose P1 folds

P1 wins -1
𝑐 − 𝐵 + 1 1 − 𝑐 = −1
Which yields:
𝑐 = 𝐵/(𝐵 + 2)
La Relance

We knew the optimal strategy for P2 is to always bet
when Y>c. Assume the optimal strategy for player I is:
Bet when X>c (No reason to fold when X>c since P2 always
folds when Y<c)
 Bet with a certain probability p when X<c (Bluff)


Now PI should choose p so that P2’s decision is
indifferent:
𝑝 ≝ 𝑃(𝑃1 𝑏𝑒𝑡𝑠|𝑋 < 𝑐)
Using Bayes’ theorem:
𝑐𝑝
𝑃 𝑋 < 𝑐 𝑃1 𝑏𝑒𝑡𝑠 =
𝑐𝑝 + (1 − 𝑐)
La Relance

Consider P2’s Decision at Y=c:
 If
P2 calls with Y=c, he/she wins pot if X<c and loses if
X>c:
𝑉𝑎𝑙𝑢𝑒 𝑃2 = 𝐵 + 1 × 𝑃 𝑋 < 𝑐 𝑃1 𝑏𝑒𝑡𝑠 − 𝑃(𝑋 > 𝑐|𝑃1 𝑏𝑒𝑡𝑠)
 If
P2 folds, Value for P2 is -1.
 Solve the equation:
𝐵 + 1 × 𝑃 𝑋 < 𝑐 𝑃1 𝑏𝑒𝑡𝑠 − 𝑃(𝑋 > 𝑐|𝑃1 𝑏𝑒𝑡𝑠) = −1
We get:
𝑝 = 1 − 𝑐 = 2/(𝐵 + 2)
La Relance

Now we can compute the value of the game as we
did in AKQ game:
𝑉𝑃1

𝐵2
𝐵 =−
(𝐵 + 2)2
Result shows the game favors P2.
La Relance

When to bluff if P1 gets a number X<c?
P1 bluffs with c2<X<c, (best hand not
betting), bets with X>c and folds with X<c2.
 Why?
 Intuitively,
 If
P2 is playing with the optimal strategy, how to choose
when to bluff is not relevant.
 This penalizes when P2 is not following the optimal strategy.
La Relance

What if player / opponent is suboptimal?

Assumed Strategy
 player
1 should always bet if X > m, fold otherwise
 player 2 should always call if Y > n, fold otherwise,
Also call if n > m is known (why?)


Assume decisions are not random beyond cards
dealt
Alternate Derivations Follow
La Relance
La Relance (Player 2 strategy)
La Relance (Player 2 strategy)


What can you infer from the properties of this
function?
What if m ≈ 0? What if m ≈ 1?
La Relance (Player 1 response)

Player 1 does not have a good response strategy
(why?)
La Relance (Player 1 Strategy)


Let’s assume player 2 doesn’t always bet when n >
m
This function is always increasing, is zero at n = β /
(β + 2)
 What
should player 1 do?
La Relance (Player 1 Strategy)

If n is large enough, P1 should always bet (why?)
If n is small however, bet when m >

What if n = β / (β + 2) exactly?

Von Neumann

Betting tree:
Von Neumann
Von Neumann

Since P1 can check,
 now
he gets positive value out of the game
 P1 now bluff with the worst hand. Why?
 On
the bluff part, it’s irrelevant to choose which section of
(0,a) to use if P2 calls (P2 calls only when Y>c)
 On the check part, it’s relevant because results are
compared right away.
Von Neumann


Nash’s equilibrium:
Three key points:
P1’s view: P2 should be indifferent between folding/calling
with a hand of Y=c
𝐵 + 1 𝑎 − (𝐵 + 1) 1 − 𝑏 = −1
 P2’s view: P1 should be indifferent between checking and
betting with X=a
𝑐− 𝐵+1 1−𝑐 =𝑎
 P2’s view: P1 should be indifferent between checking and
betting with X=b
𝑐+ 𝐵+1 𝑏−𝑐 − 𝐵+1 1−𝑏 =𝑏

Von Neumann

What if player / opponent is suboptimal?

Assumed Strategy
 Player
1 Bet if X < a or X > b, Check otherwise
 Player 2 Call if Y > c, fold otherwise
 If c is known, Player 1 wants to keep a < c and b > c
Von Neumann
Von Neumann
Von Neumann (Player 1 Strategy)

Find the maximum of the payoff function

a=

b=

What can we conclude here?
Von Neumann (Player 2 Response)

Player 2 does not have a good response strategy
Von Neumann (Player 2 Strategy)


This analysis is very similar to Borel Poker’s player 1
strategy, won’t go in depth here…
c=
Bellman & Blackwell

Bet tree
Where 0 ≤ 𝐵1 ≤ 𝐵2


Borel: B1= B2
Von Neumann: B1= 0
Bellman & Blackwell
mL
Fold
mH
Low B
b1
High B
b3
Low B
b2
High B
Bellman & Blackwell

Or
if
Where
La Relance: Non-identical Distribution



Still follows the similar pattern
Where F and G are distributions of P1 and P2, c is
still the threshold point for P2. π is still the
probability that P1 bets when he has X<c.
What if
?
La Relance:
(negative) Dependent hands



X and Y conforms to FGM distribution
Marginal distributions are still uniform.
𝜃 is correlation factor. 𝜃 < 0 means negative
correlation.
La Relance:
(negative) Dependent hands

Player 1 bets when X > l
 P(Y

< c | X = l) = B / (B + 2)
Player 2 bets when Y > c
 (2*B

+ 2)*P(X > c|Y = c) = (B + 2)*P(X > l|Y = c)
Game Value:
 P(X
> Y) – P(Y < X)
 + B * [ P(c < Y < X) – P(l < X < Y AND Y > c) ]
 + 2 * [ P(X < Y < c AND X > l) – P(Y < X < l) ]
Von Neumann:
Non-identical Distribution

Also similar to before (just substitute the distribution
functions)
a
| (B + 2) * G(c) = 2 * G(a) + B
 b | 2 * G(b) = G(c) + 1
 c | (B + 2) * F(a) = B * (1 – F(b))
Von Neumann:
(negative) Dependent hands

Player 2 Optimal Strategy:

Player 1 Optimal Strategy:
Discussion / Thoughts / Questions

Is this a good model for poker?