Game Theory

Section Seven: Game Theory

Chapter Fourteen: Game Theory

GOALS OF THIS CHAPTER

-see a brief introduction of probability using the Roulette layout

-explain the idea of expected value and house edge

-how to “count cards” using the Hi-Lo System

-introduce 2x2 zero-sum games and strategy pairs

-discuss player counterstrategies

-utilize the Maximin Theorem to find an optimalgame strategies

PROBABILITY

Let’s talk about probability through the Roulette Wheel. Here is the layout:

-we can bet on numbers, colours, rows, columns and more!

-each type of bet has a different kind of “payout” (the amount of money you win)

-to keep it easy, let’s just bet $1 at a time

PROBABILITY

Here is the Roulette wheel:

-this helps us decide when we win our bet or lose our bet

-there are 37 numbers (0-36), some are black some are red (zero is neither red or black)

-a “croupier” will spin a ball and the ball lands on a certain number

-everything is completely random

PROBABILITY

Let’s place a $1 bet on a single number:

How many ways can I win?

How many ways can I lose?

PROBABILITY

Let’s place a $1 bet on a single number:

The probability (or chance) that I win?

The probability (or chance) that I lose?

PROBABILITY

Let’s place a $1 bet on a colour:

How many ways can I win?

How many ways can I lose?

PROBABILITY

Let’s place a $1 bet on a colour:

The probability (or chance) that I win?

The probability (or chance) that I lose?

PROBABILITY

Here are some definitions I will be using throughout:

A probability experiment is a situation involving chance that leads to results called basic outcomes. Basic outcomes are a result of a single trial of the experiment. We will denote a probability experiment with a Greek letter (omega): Ω.

-in our Roulette example, the probability experiment is…

-in our Roulette example, the basic outcomes are…

PROBABILITY


An event is one (or more) basic outcomes of a probability experiment. We generally denote an even with a capital X.

-in our Roulette example, an event is…

-in our Roulette example, another event is…

PROBABILITY


The probability of an event X occurring is found by dividing the number of ways the even can occur by the total number of basic outcomes. We denote the probability of an even occurring as P(X).

-the total number of basic outcomes of the Roulette probability experiment is…

EXPECTED VALUE AND HOUSE EDGE

Now that we know probabilities, we can figure out the question everyone asks me:WHY DOES THE CASINO ALWAYS WIN?

-we need to know the payout for a single number bet: if the number 26 comes up, we win $35

-if the number does not come up, we lose our $1 bet


WHY DOES THE CASINO ALWAYS WIN?

The probability I win times how much I will win?

The probability I lose times how much I will lose?


WHY DOES THE CASINO ALWAYS WIN?

What number do we get if we add these two fractions together?

In gambling lingo, we call this the house edge. It tells a player what percentage of money he or she will pay the casino in the long run.


What about a different bet?

-we need to know the payout for a single colour bet: if the colour black comes up, we win $1

-if the colour black does not come up, we lose our $1 bet



The probability I win times how much I will win?

The probability I lose times how much I will lose?


What number do we get if we add these two fractions together?

The casino ALWAYS has an advantage over the player and the reason is all a matter of mathematics!


Another definition:

A probability experiment Ω has a special value associated to it. This value is called the expected value and is denoted as E(Ω). This number is an “average value” of the experiment if it were to be repeated many times. We can calculate the expected value by multiplying all basic outcomes by their respective probabilities:

E(Ω) = P(X1)X1 + P(X2)X2 + … + P(Xn)Xn

-in our Roulette example, the house edge is a special type of expected value


First, I want to explain why card counting came about and how it is tied to probability and expected value:

-blackjack is a game where you play against the dealer; two initial cards are dealt and additional cards are drawn to better the hand; whoever is closer to 21 without going over is the winner

-aces are worth either 1 or 11; face cards are worth 10

-it was noticed that as high cards came out of the shoethe hands that came afterwards favoured the dealerwinning

CARD COUNTING AND BLACKJACK

-the type of count I know is called the Hi-Lo count and was introduced in 1963 by Harvey Dubner

First, I want to explain why card counting came about and how it is tied to probability and expected value:

-as low cards came out of the shoe the hands that came afterwards favoured the player winning


-blackjack generally has a house edge of about 0.5%; this means out of every $100 I spend at the table, I expect to lose 5 cents

-the big question at the time was “is it possible to keep track of the cards in such a way so that the player will gain a slight advantage?”

-as it turns out, the answer is YES! using the Hi-Lo count will shift the house edge to about -1% (this means the player now expects to WIN money!)

So how do we do it?

-each card in the deck gets assigned a special value:


Aces and Face Cards(the “Hi” cards)

Value = -1

So how do we do it?



Cards 2 Through 6(the “Lo” cards)

Value = +1

So how do we do it?



Cards 7 Through 9(neutral cards)

Value = 0

So how do we do it?

-as the cards come out, we keep a “running count” of the values of the cards


+1 -1 -1 +1 0

Running Count:

1 0 0 1 0

So how do we do it?

-if at the end of the round, the running count is positive, we bet more (this means lots of small cards have been dealt and we expect more high cards to come out)

-if at the end of the round, the running count is negative, we would bet less (a lot of high cards have come out and we expect to see more low cards come out next)


What is the running count at the end of this round?


Dealer

Player 1 Player 2

Player 3

2x2 ZERO SUM GAMES

Here are some definitions that we need for the rest of the chapter:

A game is a situation involving participants (called players) that we can represent with a matrix.

Player A chooses rows of the matrix with some probability and Player B chooses columns of the matrix with some probability. At the intersection of the row and column chosen is a payoff (the entries in the matrix represent a gain or loss of a resource).

2x2 ZERO SUM GAMES

Ex. 1 – Rock Paper Scissors

Let’s set up a rock paper scissors game. If a player loses, he or she has to pay $1 to the other player. If a player wins, he or she gains $1 from the other player. A tie results in no money exchange. Remember that paper beats rock, rock beats scissors and scissors beats paper.

2x2 ZERO SUM GAMES

We call a game a zero-sum game if the total wins and losses of both players totals zero. For this course, we will only focus on 2x2 zero-sum games.

The rock, paper and scissors example is a zero-sum game. It is not a 2x2 game though (it’s 3x3). A 2x2 game will be represented with a 2x2 matrix.

A strategy pair is a pair of numbers (a,b) that satisfy 0 ≤ a ≤ 1, 0 ≤ b ≤ 1 and a + b = 1.

We generally use the pair (1-p,p) for Player A and the pair (1-q,q) for Player B. Here, p denotes the probability of Player A of choosing the second row and q denotes the probability of Player B choosing the second column.

2x2 ZERO SUM GAMES

The general form of a 2x2 zero-sum game is:

We can calculate the various payoff probabilities:

Finally, we can calculate the expected value of the game:

2x2 ZERO SUM GAMES

Ex. 2 – Expected Value Calculation

Bob (Player A) and Sue (Player B) play a 2x2 zero-sum game with the following representation:

5 0

-1 2

If the strategy pairs of Bob and Sue are (0.2,0.8) and (0.3,0.7), respectively, calculate the expected value of Bob’s game.

COUNTER STRATEGY

Ex. 3 – Counter Strategy

Bob (Player A) and Sue (Player B) play a 2x2 zero-sum game with the following representation:

5 0

-1 2

If only the strategy pair of Sue is known (0.3,0.7), how should Bob play his game to maximize his strategy?

THE MAXIMIN THEOREM

Counter strategy is good for when we know how Player B will play the game – it will ensure the best possible expected value.

What do we do if we don’t know how often Player B will choose column 1 or column 2? In this section we will develop a way to play the game when we do not know Player B’s strategy pair.

We will assume that Player B has an unknown strategy pair and that Player B is always trying to minimize the expected value of Player A.

THE MAXIMIN THEOREM

First, we will consider two different expected values. The first comes from using the strategy pair (0,1) for Player B and the second comes from using the strategy pair (1,0) for Player B.

THE MAXIMIN THEOREM

Since Player B wants to minimize Player A’s winnings as much as possible, we create a third expected value called Emin. This expected value is found by taking the minimum of both E1 and E2.

If we plot Emin as a function of p, we can find the best way to maximize Player A’s winnings.

Ex. 4 – Penny Matching Example

Suppose that Player A and Player B play a game where they decide to play a penny heads-up or tails-up. If both pennies show the same side, Player B gets both pennies and if the pennies show different sides, Player A gets the two pennies. Find a matrix representation for the following 2x2 zero-sum game and maximize Player A’s expected value.

THE MAXIMIN THEOREM

Thm. 5 – (Maximin Theorem) If (p,E(Ω)) is the highest point on the graph of Emin(Ω), then (1-p,p) is a maximin strategy pair for Player A and E(Ω) is Player’ A’s maximin expectation. If Player A uses the maximin strategy pair (1-p,p), then Player A expects to win E(Ω) on each play of the game.

THE MAXIMIN THEOREM

Ex. 5 – Maximin Question

Find the maximin strategy pair and the maximin expectation of Player A for the following 2x2 zero-sum game:

0.8 1

0.9 0.5

THE MAXIMIN THEOREM

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Game Theory

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