ORF 307: Lecture 12. Linear Programming: Chapter 11: Game Theory

Transcription

1 ORF 307: Lecture 12 Linear Programming: Chapter 11: Game Theory Robert J. Vanderbei April 3, 2018 Slides last edited on April 3, rvdb

2 Game Theory John Nash = A Beautiful Mind 1

3 Game Theory The other John Nash 2

4 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 dulls Scissors Paper covers Rock Scissors cuts Paper Check out Sam Kass version: Rock, Paper, Scissors, Lizard, Spock It was featured on The Big Bang Theory. 3

5 Payoff Matrix Payoffs are from row player to column player: A = R P S R P S Note: Any deterministic strategy employed by either player can be defeated systematically by the other player. 4

6 Two-Person Zero-Sum Games Given: m n matrix A. Row player selects a strategy i {1,..., m}. Column player selects a strategy j {1,..., n}. Row player pays column player a ij dollars. Note: The rows of A represent deterministic strategies for row player, while columns of A represent deterministic strategies for column player. Deterministic strategies can be (and usually are) bad. 5

7 Randomized Strategies. Suppose row player picks i with probability y i. Suppose column player picks j with probability x j. Throughout, x = [ x 1 x 2 ] T x n and y = [ y 1 y 2 ] T y m will denote stochastic vectors: x j 0, j = 1, 2,..., n x j = 1 j y i 0, i = 1, 2,..., m y i = 1 i If row player uses random strategy y and column player uses x, then expected payoff from row player to column player is y i a ij x j = y T Ax i j 6

8 Column Player s Analysis Suppose column player were to adopt strategy x. Then, row player s best defense is to use strategy y that minimizes y T Ax: min y y T Ax And so column player should choose that x which maximizes these possibilities: max x min y y T Ax 7

9 Quiz What s the solution to this problem: minimize 3y 1 + 6y 2 + 2y y 4 + 7y 5 subject to: y 1 + y 2 + y 3 + y 4 + y 5 = 1 y i 0, i = 1, 2, 3, 4, 5 8

10 Solving Max-Min Problems as LPs Inner optimization is easy: min y y T Ax = min i e T i Ax (e i denotes the vector that s all zeros except for a one in the i-th position that is, deterministic strategy i). Note: Reduced a minimization over a continuum to one over a finite set. We have: max (min i x j = 1, j e T i Ax) x j 0, j = 1, 2,..., n. 9

11 Reduction to a Linear Programming Problem Introduce a scalar variable v representing the value of the inner minimization: max v v e T i Ax, i = 1, 2,..., m, x j = 1, j x j 0, j = 1, 2,..., n. Writing in pure matrix-vector notation: max v ve Ax 0 e T x = 1 x 0 (e without a subscript denotes the vector of all ones). 10

12 Finally, in Block Matrix Form [ ] T [ ] 0 x max 1 v [ A e e T 0 ] [ x v ] = [ 0 1 ] x 0 v free 11

13 Row Player s Perspective Similarly, row player seeks y attaining: min y max x y T Ax which is equivalent to: min u ue A T y 0 e T y = 1 y 0 12

14 Row Player s Problem in Block-Matrix Form [ ] T [ ] 0 y min 1 u [ A T e e T 0 ] [ y u ] = [ 0 1 ] y 0 u free Note: Row player s problem is dual to column player s: [ 0 max 1 ] T [ x v ] [ 0 min 1 ] T [ y u ] [ A e e T 0 ] [ x v ] = [ 0 1 ] [ A T e e T 0 ] [ y u ] = [ 0 1 ] x 0 v free y 0 u free 13

15 MiniMax Theorem Theorem. Let x denote column player s solution to her max min problem. Let y denote row player s solution to his min max problem. Then max x Proof. From Strong Duality Theorem, we have Also, y T Ax = min y y T Ax. u = v. v u = min i = max j e T i Ax = min y y T Ae j = max x y T Ax y T Ax QED 14

16 AMPL Model set ROWS; set COLS; param A {ROWS,COLS} default 0; var x{cols} >= 0; var v; maximize zot: v; subject to ineqs {i in ROWS}: sum{j in COLS} -A[i,j] * x[j] + v <= 0; subject to equal: sum{j in COLS} x[j] = 1; 15

17 AMPL Data data; set ROWS := P S R; set COLS := P S R; param A: P S R:= P S R ; solve; printf {j in COLS}: " %3s %10.7f \n", j, 102*x[j]; printf {i in ROWS}: " %3s %10.7f \n", i, 102*ineqs[i]; printf: "Value = %10.7f \n", 102*v; 16

18 AMPL Output ampl gamethy.mod LOQO: optimal solution (12 iterations) primal objective dual objective P S R P S R Value =

19 Dual of Problems in General Form (Review) Consider: max c T x Ax = b x 0 Rewrite equality constraints as pairs of inequalities: max c T x Ax b Ax b x 0 Put into block-matrix form: [ max ct ] x A x [ A x 0 b b ] Dual is: [ ] T [ ] b y + min b y [ ] [ ] A T A T y + y Which is equivalent to: c y +, y 0 min b T (y + y ) A T (y + y ) c y +, y 0 Finally, letting y = y + y, we get min b T y A T y c y free. 18

20 Summary 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. 19

21 A Real-World Example As before, we can let let R t,j = S j (t)/s j (t 1) and view R as a payoff matrix in a game 20 between Fate and the Investor. The Ultra-Conservative Investor Consider again some historical investment data (S j (t)): XLU-utilities XLB-materials XLI-industrials XLV-healthcare XLF-financial XLE-energy MDY-midcap XLK-technology XLY-discretionary XLP-staples QQQ SPY-S&P500 XLY QQQ XLV MDY XLK 2.5 XLI SPY XLP XLB Share Price 2 XLF XLU XLE Date

22 Fate s Conspiracy The columns represent pure strategies for our conservative investor. The rows represent how history might repeat itself. Of course, for tomorrow, Fate won t just repeat a previous day s outcome but, rather, will present some mixture of these previous days. 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: XLP 98.4 XLU 1.6 Mean Old Fate s Mix: = Black Monday (2011) The value of the game is the investor s expected return, 96.2%, which is actually a loss of 3.8%. The data can be download from here: Here, xlu is just one of the funds of interest. 21

23 Starting From To Ignore Black Monday (2011) XLU-utilities XLB-materials XLI-industrials XLV-healthcare XLF-financial XLE-energy MDY-midcap XLK-technology XLY-discretionary XLP-staples QQQ SPY-S&P500 XLV XLY XLF QQQ Share Price MDY XLI SPY XLK XLP XLB XLU 1.2 XLE Date 22

24 Fate s Conspiracy Investor s Optimal Asset Mix: XLK 75.5 XLV 15.9 XLU 6.2 XLB 2.2 XLI 0.2 Mean Old Fate s Mix: The value of the game is the investor s expected return, 97.7%, which is actually a loss of 2.3%. 23

25 Giving Fate Fewer Options Thousands seemed unfair How about XLU-utilities XLB-materials XLI-industrials XLV-healthcare XLF-financial XLE-energy MDY-midcap XLK-technology XLY-discretionary XLP-staples QQQ SPY-S&P500 XLV 1 XLU MDY Share Price 0.98 XLE XLY XLP XLF SPY 0.96 QQQ XLI XLK 0.94 XLB Day in March

26 Fate s Conspiracy Investor s Optimal Asset Mix: MDY 83.7 XLE 13.2 XLF 3.2 Mean Old Fate s Mix: The value of the game is the investor s expected return, 98.7%, which is actually a loss of 1.3%. 25