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Linear Optimization: A Geometric Inquiry Course

Tien Chih

Section 5.3 Games of Chance

For games with random components, such as gambling, we can still apply our techniques. This requires some work, and we explore how to do so.

Activity 5.3.1.

Consider the following game. Rowan and Colleen place $25 into a pot. Then they are dealt either a Jack, Queen or King at random. This deck only has those three cards. Whoever has the highest card takes the pot.
Rowan has the option of raising by $10 or folding. If he folds, he loses $25. Otherwise Colleen then can either fold or call. If they fold, she loses $25, if she calls, she puts $10 into the pot.
Note that if Rowan folds, even if Colleen had planned on folding, she would win the $25.

(a)

What would best describe possible choices of strategy for Rowan and Colleen?
  1. The cards Jack, Queen, King.
  2. Whether to Fold/Raise for Rowan, whether to Fold/Call for Colleen.
  3. Whether to Fold/Raise for Rowan depending on the card he is dealt, whether to Fold/Call for Colleen, depending on the card she is dealt.

(b)

List the possible pairs of Rowan/Colleen hands.

(c)

Let’s say Rowan raises on a Jack, folds on a Queen, and raises on a King, denoted as RFR. Let’s saw Colleen folds on a Jack, and calls on a Queen of King, denoted FCC.
If both players are committed to these strategies, what are Rowans expected net winnings? (Note that all the above hand pairs are equally likely, what are Rowan’s net winnings in each case?)

(d)

If Rowan’s strategy is (for some reason) FFF, what are Rowans net winnings?

(e)

If Colleen’s strategy is FFF, what are Rowans net winnings? (These may be different for each of Rowan’s choice of strategy.)

(f)

Without computing the entire payoff matrix, are there any obviously poor strategies for Rowan or Colleen?

(g)

Fill out the remainder of this payoff matrix, where the entries are expected values.
FFF FFC FCF CFF FCC CFC CCF CCC
FFF ? ? ? ? ? ? ? ?
FFR ? \(-25/3\) \(-20/3\) \(-20/3\) \(-20/3\) \(-20/3\) \(-5\) \(-5\)
FRF ? \(-55/3\) \(-25/3\) \(-20/3\) -55/3 \(-50/3\) \(-20/3\) \(-50/3\)
RFF ? \(-55/3\) \(-55/3\) \(-25/3\) \(-85/3\) \(-55/3\) \(-55/3\) -85/3
FRR ? \(-5/3\) \(10\) \(35/3\) \(0\) \(5/3\) \(40/3\) \(10/3\)
RFR ? \(-5/3\) \(0\) \(10\) ? \(0\) \(5/3\) \(-25/3\)
RRF ? \(-35/3\) \(-5/3\) ? \(-65/3\) \(-10\) \(0\) \(-20\)
RRR ? \(5\) \(50/3\) \(85/3\) \(-10/3\) \(25/3\) \(20\) ?

(h)

After dominating what does this table reduce to?
??? ???
??? ? ?
??? ? ?

(i)

Solve for the optimal strategies using the method of your choice.

(j)

Who does the game favor and by how much?

(k)

If Rowan is dealt a Jack, what is his optimal strategy (as a pair of probabilities to Raise or Fold). Queen? King?

(l)

If Colleen is dealt a Jack, what is her optimal strategy. Queen? King?

Activity 5.3.2.

We introduce a second game here. Each player places $\(b\) into the pot. Then the each secretly flip a coin. We consider heads “greater” than tails.
Rowan then has the options of passing or betting. If he passes, then both players reveal their coin and the higher value wins. If they are the same, players split the pot evenly, and both players net wins/losses are $0. His other option is to bet, in which case he adds $\(r\) to the pot.
Then Colleen has a choice as well, to fold or call. If she folds, then Rowan nets the $\(b\text{.}\) Otherwise, she calls, and also adds $\(r\) to the pot and both coins are revealed.

(a)

Suppose that Rowan will stick to the strategy of RP (raise on head, pass on tails) and Colleen choses CC (call on both heads or tails). What are Rowan’s expected winnings in this case?

(b)

Fill out the payoff matrix for this game.
FF FC CF CC
PP ? \(0\) \(0\) ?
PR ? \(b/2\) \((b-r)/4\) ?
RP \(b\) ? \(0\) ?
RR \(b\) ? ? \(0\)

(c)

It’s not possible to determine all the domination without knowing what \(b, r\) are. However, knowing \(b,r>0\text{,}\) dominate as much as possible
?? ??
?? ? ?
?? ? ?

(d)

If \(b\leq r\text{,}\) use domination to find the optimal pure strategy for both players.

(e)

If \(b>r\text{,}\) use linear optimization methods to find the optimal mixed strategies for both players.
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