1/7/12 Game Theory Lecture 13 Zero‐sum games •  In zero‐sum games, payoffs in each cell sum up to zero •  Movement diagram 1 1/7/12 Zero‐sum games •  Minimax = maximin = value of the game •  The game may have mulFple saddle points Zero‐sum games •  Or it may have no saddle points •  To find the value of such game, consider mixed strategies 2 1/7/12 Zero‐sum games •  If there is more strategies, you don’t know which one will be part of opFmal mixed strategy. •  Let Column mixed strategy be (x,1‐x) •  Then Raw will try to maximize Zero‐sum games •  Column will try to choose x to minimize the upper envelope 3 1/7/12 Zero‐sum games •  Tranform into Linear Programming Fishing on Jamaica •  In the fiVies, Davenport studied a village of 200 people on the south shore of Jamaica, whose inhabitants made their living by fishing. 4 1/7/12 •  Twenty‐six fishing crews in sailing, dugout canoes fish this area [fishing grounds extend outward from shore about 22 miles] by se9ng fish pots, which are drawn and reset, weather and sea permi9ng, on three regular fishing days each week … The fishing grounds are divided into inside and outside banks. The inside banks lie from 5‐15 miles offshore, while the outside banks all lie beyond … Because of special underwater contours and the locaEon of one prominent headland, very strong currents set across the outside banks at frequent intervals … These currents are not related in any apparent way to weather and sea condiEons of the local region. The inside banks are almost fully protected from the currents. [Davenport 1960] Jamaica on a map 5 1/7/12 Strategies •  There were 26 wooden canoes. The captains of the canoes might adopt 3 fishing strategies: –  IN – put all pots on the inside banks –  OUT – put all pots on the outside banks –  IN‐OUT) – put some pots on the inside banks, some pots on the outside Advantages and disadvantages •  It takes Fme to reach outside banks, so those who use OUT or IN‐OUT can set fewer pots •  When the current is running, it is harmful to outside pots: marks are dragged away, pots may be smashed while moving, changes in temeperature may kill fish inside the pots, etc. •  The outside banks produce higher quality fish, both in variaFes and in size. If many outside fish are available, they may drive the inside fish off the market. •  The OUT and IN‐OUT strategies require befer canoes. Their captains dominate the sport of canoe racing, which is presFgious and offers large rewards. 6 1/7/12 CollecFng data •  Davenport collected the data concerning the fishermen average monthly profit depending on the fishing strategies they used to adopt. Fishermen\Current FLOW NO FLOW IN 17,3 11,5 OUT ‐4,4 20,6 IN‐OUT 5,2 17,0 Zero‐sum game ??? •  The problem of “vicious current”; p – current’s strategy •  Fishermen problem; q – fishermen strategy 7 1/7/12 Graphical soluFon of the current’s problem 21 19 17 SoluFon: p=0.31 15 IN 13 OUT 11 IN‐OUT 9 Mixed strategy of the current 7 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Maximin and minimax objective function
Maximize
13,31
Expected payoff of the current when
FLOW
NO FLOW
probabilities
13,31
13,31
1,00
minimize
Expected payoff from strategy:
IN
OUT
IN_OUT
probabilities
objective
function
13,31
13,31
12,79
13,31
1,00
q1
Fishers' mixed strategy
q2
q3
0,67
0,00
0,33
>=
>=
=
13,31
13,31
1,00
Mixed strategy of the current
p
1-p
0,31
0,69
<=
<=
<=
=
13,31
13,31
13,31
1,00
8 1/7/12 Forecast and observaFon Game theory predicts ObservaMon shows •  No fishermen risks fishing outside •  Strategy 67% IN, 33% IN‐
OUT [Payoff: 13.31] •  OpFmal current’s strategy 31% FLOW, 69% NO FLOW •  No fishermen risks fishing outside •  Strategy 69% IN, 31% IN‐
OUT [Payoff: 13.38] •  Current’s „strategy”: 25% FLOW, 75% NO FLOW The similarity is striking Davenport’s finding went unchallenged for several years UnFl … Current is not vicious •  Kozelka 1969 and Read, Read 1970 pointed out a serious flaw: –  The current is not a reasoning enFty and cannot adjust to fishermen changing their strategies. –  Hence fishermen should use Expected Value principle: •  Expected payoff of the fishermen: –  IN: 0.25 x 17.3 + 0.75 x 11.5 = 12.95 –  OUT: 0.25 x (‐4.4) + 0.75 x 20.6 = 14.35 –  IN‐OUT: 0.25 x 5.2 + 0.75 x 17.0 = 14.05 •  Hence, all of the fishermen should fish OUTside. •  Maybe, they are not well adapted aVer all 9 1/7/12 Current may be vicious aVer all •  The current does not reason, but it is very risky to fish outside. •  Even if the current runs 25% of the Fme ON AVERAGE, it might run considerably more or less in the short run of a year. •  Suppose one year it ran 35% of the Fme. Expected payoffs: –  IN: 0.35 x 17.3 + 0.65 x 11.5 = 13.53 –  OUT: 0.35 x (‐4.4) + 0.65 x 11.5 = 11.85 –  IN‐OUT: 0.35 x 5.2 + 0.65 x 17.0 = 12.87. •  By treaFng the current as their opponent, fishermen GUARANTEE themselves payoff of at least 13.31. •  Fishermen pay 1.05 pounds as insurance premium OpFmal Actual OUT Actual (25%) 13.3125 13.291 14.35 Vicious (31%) 13.3125 13.31164 12.85 35% 13.3125 13.3254 11.85 •  Consider the following game Mr Raw
A
B
Mrs Column
A
B
(2,6) (10,5)
(4,8) (0,0)
10 1/7/12 Security level •  Mr Raw’s game is a zero‐sum game with Mr Raw’s payoffs unchaged A
Mr Raw
B
Mrs Column
A
B
(2,6) (10,5)
(4,8) (0,0)
X
Mr Raw
A
B
A
B
(2,-2) (10,-10)
(4,-4) (0,0)
•  Mr Raw’s security level is the value of Mr Raw’s game: –  No saddle point, so mixed strategy –  x=5/6, expected payoff=10/3=value of Mr Raw’s game. Security level •  Mrs Column’s game is a zero‐sum game with Mrs Column’s payoffs unchaged Mr Raw
A
B
Mrs Column
A
B
(2,6) (10,5)
(4,8) (0,0)
X
A
B
Mrs Column
A
B
(-6,6) (-5,5)
(-8,8) (0,0)
•  Mrs Column’s security level is the value of Mrs Column’s game: –  There is a saddle point 6 –  So 6=value of Mrs Column’s game. 11 1/7/12 Nash arbitraFon Scheme •  Up to this point, no cooperaFon btw. players •  Now different approach: what is a reasonable outcome to the game? •  Take again our previous game Mrs Column
A
B
A (2,6) (10,5)
Mr Raw
B (4,8) (0,0)
•  Egalitarian proposal: choose the outcome with the largest total payoff. Then split it equally. (15/2 for each) •  Two major flaws of this approach: –  Payoffs are uFliFes. They cannot be meaningfully added or transferred across individuals. –  Neglects the asymmetries of strategic posiFon in the game. (Mrs. Column posiFon in the game above is strong) Nash arbitraFon scheme •  Challenge: to find a method of arbitraFng games which: –  Does not involve illegiFmate manipulaFon of uFliFes –  Does take into account strategic ineqaliFes –  Has a claim to fairness •  First good idea: von Neumann, Morgenstern (1944): any reasonable soluFon to a non‐zero‐sum game should be: –  Pareto‐opMmal –  At or above the security level for both players •  The set of such outcomes (pure or mixed) is called the negoMaMon set of the game. 12 1/7/12 Payoff polygon and Pareto‐efficient points Mrs Column’s payoff (4,8) Pareto‐efficient fronMer (2,6) (10,5) Mr Raw’s payoff (0,0) Status Quo and negoFaFon set Mrs Column’s payoff (4,8) NegoMaMon set 6 (2,6) SQ (10,5) Mr Raw’s payoff (0,0) 10/3 13 1/7/12 Nash 1950 arbitraFon scheme •  What point in the negoFaFon set do we choose? –  Nash arbitraFon scheme: •  Axioms: 1.  RaMonality: The soluFon lies in the negoFaFon set 2.  Linear invariance: If we linearly transform all the uFliFes, the soluFons will also be linearly transformed 3.  Symmetry: If the polygon happens to be symmetric about the line of slope +1 through SQ, then the soluFon should be on this line as well 4.  Independence of Irrelevant AlternaMves: Suppose N is the soluFon point for a polygon P with status quo point SQ. Suppose Q is another polygon which contains both SQ and N, and is totally contained in P. Then N should also be the soluFon point for Q with status quo point SQ. Independence of Irrelevant AlternaFves graphically N SQ Suppose N is the soluFon point for a polygon P with status quo point SQ. Suppose Q is another polygon which contains both SQ and N, and is totally contained in P. Then N should also be the soluFon point for Q with status quo point SQ. 14 1/7/12 Nash 1950 Theorem •  Theorem: There is one and only one arbitraMon scheme which saFsfies Axioms 1‐4. It is this: if SQ=(x0,y0), then the arbitrated soluFon point N is the point (x,y) in the polygon with x ≥ x0, and y ≥ y0, which maximizes the product (x‐x0)(y‐y0). Nash soluFon – our example •  In our example Nash soluFon is (5 2/3, 7 1/6) Mrs Column’s payoff (4,8) 6 (2,6) (5 2/3, 7 1/6) SQ (10,5) Mr Raw’s payoff (0,0) 10/3 15 1/7/12 ApplicaFon •  The management of a factory is negoFaFng a new contract with the union represenFng its workers •  The union demands new benefits: –  One dollar per hour across‐the‐board raise (R) –  Increased pension benefits (P) •  Managements demands concessions: –  Eliminate the 10:00 a.m. coffee break (C) –  Automate one of the assembly checkpoints (reducFon necessary) (A) •  You have been called as an arbitrator. 16