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APPENDIX B

APPENDIX B. APPENDIX B A PROCEDURE FOR GENERATING AN EQUILIBRIUM POINT FOR 2-PERSON GAMES (That sometimes works!). Basic Idea. Player II speaking : "I'll design my strategy, call it y* such that the expected payoff to Player I will be constant regardless of what she will do!

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APPENDIX B

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  1. APPENDIX B

  2. APPENDIX BA PROCEDURE FOR GENERATING AN EQUILIBRIUM POINT FOR 2-PERSON GAMES(That sometimes works!)

  3. Basic Idea • Player II speaking: • "I'll design my strategy, call it y* such that the expected payoff to Player I will be constant regardless of what she will do! • This will produce a stable situation for me!!!!!! • Thus, to compute y* I have to solve the following equation: xAy* = Constant • Any such y* will certainly satisfy xAy* ≤ x*Ay*, for all x in S, regardless of what Player I is selecting for x* (her best strategy)."

  4. Needless to say, Player I will attempt to do the same thing and construct a strategy x* such that x*By is independent of y. • How then do we solve xAy* = Constant ??? Let z = Ay* Then xAy* = xz = (x1, x2, ..., xm) (z1, z2, ..., zm)t = (x1z1 + x2z2 + ... xmzm) Now if z1 = z2 = ... = zm, this becomes xAy* = (x1 + x2 + ... xm) zm = zm= constant REGARDLESS OF WHAT x IS!! How do we use this? We can try putting all components of Ay* equal to each other.

  5. Particular case: n = m = 2 y* = (y*1, y*2 )= (y*1, 1 – y*1) = (ay*1 + b(1 – y*1), cy*1 + d(1 – y*1))

  6. d  b y *  1 a  d  ( b  c ) a  c y *  2 a  d  ( b  c ) • (z1, z2) = (ay*1 + b(1 – y*1), cy*1 + d(1 – y*1) • For z1 = z2 we obtain • ay*1 + b(1 – y*1) = cy*1 + d(1 – y*1) • So, provided a + d – (b + c) = 0 • NOTE: Beware! This y* formula is obtained from matrix A. The x* will use matrix B. since y*2 = 1 – y*1

  7. Example 1.8.2 (Continued)

  8. checking z=Ay* is constant

  9. Exercise a b c d • Derive the recipe for x* in the case n = m = 2. • Answer: If B = ( ) • x* = ((d–c)/(a+d–b–c), (a–b)/(a+d–b–c)) Example: • Find an equilibrium pair (X*, Y*) of mixed strategies for the 2-person non-zero sum game with payoff matrix • See lecture for solution.

  10. Appendix A • See lecture for discussion and examples.

  11. Summing up NON-cooperative, NON-zero-sum games

  12. Solution concepts of NON-zero-sum games • We have two concepts • Security level pairs, Equilibrium pairs • The security level idea is not really very good here, because it assumes a player is simultaneously trying to maximize their own payoff, whilst minimizing their opponents payoff. These two objectives are sometimes diametrically opposed. It is no longer true that a player can get rich only by keeping their opponent poor. • Also we know that the payoff for secuity level is ≤ that for equilibrium pair.

  13. Equilibrium pairs - more acceptable concept, but difficulties with these too. • E.g. • Only one equilibrium pair, payoff (1,1). But clearly the payoff (5,5) better for both. • No satisfactory simple notion of ‘optimal strategy’ and ‘value’ as there is with zero sum games.

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