Economics 2020b Lecture 9:
Simple Poker
A. The General Problem
Your position is weak, but your opponent doesn’t know
this.
Should you pretend to be strong?
And what about your opponent – how should they
respond to a “strong” action?
This question lies at the heart of many applications of
Game Theory, e.g., “signaling games”.
Key insight
To analyze a situation where you are weak but your
opponent doesn’t know this, the game model should
explicitly include the possibility that you are strong.
I is Weak
I is Strong
I
II
I
II
Furthermore, to facilitate analysis, the model must
specify the probability of each type (Weak of Strong).
Such a model is called a Bayesian Game.
The reason for the adjective “Bayesian” is that there are
prior probabilities about the players’ types.
This is parallel to the key assumption of Bayesian
Statistics, namely that one can start the analysis by
assessing prior probabilities.
A Nash equilibrium in a Bayesian game is often referred
to as a Bayes-Nash equilibrium.
B. Simple Poker
$20 in pot.
Two cards in a hat: Hi and Lo.
One card chosen at random.
Player I sees the card. Player II does not.
I can pass  II gets pot
or bet Add $10 to pot; then
II can fold  I gets pot
or call Add $10 to pot; then
Card checked
Hi  pot to I
Lo  pot to II
Call
Bet
Hi
II
Fold
I
½
Pass
Call
Bet
½
Lo
II
Fold
I
Pass
This is a constant-sum game.
Q. What is the value of the game for Player I? 13.3
Q. You are Player I. You see the card is Hi.
- What should you do?
Bet
- What is the value of your position?
26.67
Q. You are Player I. You see the card is Lo.
- What should you do?
Bet (bluff) with prob 1/3
- What is the value of your position?
0
Strategic Game:
. 𝟓 ∙ 𝟑𝟎 + . 𝟓 ∙ (−𝟏𝟎)
𝟐𝟎
. 𝟓 ∙ 𝟑𝟎 + . 𝟓 ∙ 𝟎
. 𝟓 ∙ 𝟐𝟎+ . 𝟓 ∙ 𝟎
. 𝟓 ∙ 𝟎+ . 𝟓 ∙ (−𝟏𝟎)
. 𝟓 ∙ 𝟎+ . 𝟓 ∙ 𝟐𝟎
𝟎
𝟎
10
20
15
10
Equilibrium strategies:
I. Hi: Bet
Lo: Bet (bluff) with prob 1/3
II. Call with prob 2/3
Equilibrium payoffs:
𝟏
𝟐
(𝟏𝟑 𝟑 , 6𝟑)
We can now go back and answer the questions
1
Q. What is the value of the game for Player I? 13 3
Q. You are Player I. You see the card is Hi.
- What should you do? Bet
2
- What is the value of your position? 26 3
Q. You are Player I. You see the card is Lo.
- What should you do? Bet with prob 1/3
- What is the value of your position? 0
Q. Player I is following his equilibrium strategy. He
has bet. What is the probability that the card is Hi?
Bet
I
Pass
Bet
I
Pass
Deriving the solution by “backwards induction”:
Player II must mix between Fold and Call. (Why?)
What then must be P(Hi) after observing Bet?
𝑝(−10) + (1 − 𝑝)30 = 0 s so 𝑝 =
What then must be P(Bet)?
.5
.5+ .5𝑏
=
1
1+𝑏
=
3
4
so 𝑏 =
1
3
Bet
I
Pass
Bet
I
Pass
3
4
Varying the parameters of the game:
Q. What if there are 10 cards in the hat, 6 of which are
Hi? (The pot is still $20, and the cost of betting or
folding is still $10.)
14
20
18
12
Prob(Hi) = .5
Prob(Hi) = .6
P(Bluff)
.33
.5
Value (start)
13.33
16
Value (Hi)
26.67
26.67
Call
Bet
Hi
II
Fold
I
.6
Pass
Call
Bet
Lo
II
Fold
I
Pass