Linear Programming Game Theory, Lecture Notes - Mathematics, Study notes of Linear Programming

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Linear Programming: Chapter 11

Game Theory

Robert J. Vanderbei

October 17, 2007

Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/∼rvdb

Rock-Paper-Scissors

A two person game. Rules. At the count of three declare one of:

Rock Paper Scissors

Winner Selection. Identical selection is a draw. Otherwise:

• Rock beats Scissors
• Paper beats Rock
• Scissors beats Paper

Payoff Matrix. Payoffs are from row player to column player:

A =

P S R

P

S

R

Note: Any deterministic strategy employed by either player can be defeated sys- tematically by the other player.

Randomized Strategies.

• Suppose rowboy picks i with probability yi.
• Suppose colgirl picks j with probability xj.

Throughout, x =

[

x 1 x 2 · · · xn

]T

and y =

[

y 1 y 2 · · · ym

]T

will denote stochastic vectors:

xj ≥ 0 , j = 1, 2 ,... , n ∑

j

xj = 1.

If rowboy uses random strategy y and colgirl uses x, then expected payoff from rowboy to colgirl is (^) ∑

i

j

yiaij xj = yT^ Ax

Colgirl’s Analysis

Suppose colgirl were to adopt strategy x.

Then, rowboy’s best defense is to use y that minimizes yT^ Ax:

min y

yT^ Ax

And so colgirl should choose that x which maximizes these possibilities:

max x

min y

yT^ Ax

Reduction to a Linear Programming Problem

Introduce a scalar variable v representing the value of the inner minimization:

max v

v ≤ eTi Ax, i = 1, 2 ,... , m, ∑

j

xj = 1 ,

xj ≥ 0 , j = 1, 2 ,... , n.

Writing in pure matrix-vector notation:

max v ve − Ax ≤ 0 eT^ x = 1 x ≥ 0

(e denotes the vector of all ones).

Finally, in Block Matrix Form

max

[

]T [

x v

]

[

−A e eT^0

] [

x v

]

[

]

x ≥ 0 v free

Rowboy’s Problem in Block-Matrix Form

min

[

]T [

y u

]

[

−AT^ e eT^0

] [

y u

]

[

]

y ≥ 0 u free

Note: Rowboy’s problem is dual to colgirl’s.

MiniMax Theorem

Let x∗^ denote colgirl’s solution to her max–min problem. Let y∗^ denote rowboy’s solution to his min–max problem. Then max x

y∗T^ Ax = min y

yT^ Ax∗.

Proof. From Strong Duality Theorem, we have

u∗^ = v∗.

Also,

v∗^ = min i

eTi Ax∗^ = min y

yT^ Ax∗

u∗^ = max j

y∗T^ Aej = max x

y∗T^ Ax

QED

AMPL Data

data; set ROWS := P S R; set COLS := P S R; param A: P S R:= P 0 1 - S -3 0 4 R 5 -6 0 ;

solve; printf {j in COLS}: " %3s %10.7f \n", j, 102x[j]; printf {i in ROWS}: " %3s %10.7f \n", i, 102ineqs[i]; printf: "Value = %10.7f \n", 102*v;

AMPL Output

ampl gamethy.mod LOQO: optimal solution (12 iterations) primal objective -0. dual objective -0. P 40. S 36. R 26. P 62. S 27. R 13. Value = -16.

Put into block-matrix form:

[^ max cT^ x A −A

]

x

[

b −b

]

x ≥ 0

Dual is:

min

[

b −b

]T [

y+ y−

]

[

AT^ −AT^

]

[

y+ y−

]

≥ c

y+, y−^ ≥ 0

Which is equivalent to:

min bT^ (y+^ − y−) AT^ (y+^ − y−) ≥ c y+, y−^ ≥ 0

Finally, letting y = y+^ − y−, we get

min bT^ y AT^ y ≥ c y free.

Moral:

• Equality constraints =⇒ free variables in dual.
• Inequality constraints =⇒ nonnegative variables in dual.

Corollary:

• Free variables =⇒ equality constraints in dual.
• Nonnegative variables =⇒ inequality constraints in dual.

Fate’s Conspiracy

The columns represent pure strategies for our conservative investor. The rows represent how history might repeat itself. Of course, for next year (1995), Fate won’t just repeat a previous year but, rather, will present some mixture of these previous years. Likewise, the investor won’t put all of her money into one asset. Instead she will put a certain fraction into each. Using this data in the game-theory ampl model, we get the following mixed- strategy percentages for Fate and for the investor.

Investor’s Optimal Asset Mix:

US 3-MONTH T-BILLS 93. NASDAQ COMPOSITE 5. EAFE 1.

Mean, old Fate’s Mix: 1992 28. 1993 7. 1994 64.

The value of the game is the investor’s expected return: 4. 10 %.