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The payoff in guessing outcome of fair coin toss is 0, no matter
what strategy you use. The doubling your bet untill you win has
an infinite extensive form with one infinite chance position (on the
left):
1/2 $1
1/4 $1
...
...
$0? ...
0 -infinity?
where ? stands for undefined payoff.
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Equilibrium makes perfect sense only for 2-player constant-sum games
(including all 1-player games).
We discuss other concepts of solution (cooperative ones) in a month or
two.
Example:
Two players,: He, She. Normal form:
2 strategies for each player
She
He
(10,1)
(0,0)
(0,0)
(0,0)
We have one equilibrium with payoff (10,1). Moreover, no player
can gain by any chamge (unilateral or joint ).
But would not she try to get more using threats?
We assume that she is a perfect player, so her objective is the
maximize her payoff. He is also a perfect palayer, so how much
should be side payment,
if it is possible?
Another example.
2 strategies for each player
She
He
(10,0
(0,0)
(0,0)
(0,0)
Now there are 3 equilibria, so what is the solution?
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A strategy for a player in extensive form = a choice of a move in every
player's info set.
In case of perfect info, info set = position.
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Blackjack
In a movie:
The bad guy draws at 20 (he can see the top card ace in deck with his
classes) with comment "I like to live dangerously".
The good guy stands at 10 with comment " I also like to live
dangerously"
My comment: Why the bad guy need to do bad things while he can
make lots of money with his glasses without killing anybody?
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Example of Roullette
Coupon: Face value $5. Use at least $5 of your money with coupon.
Coupon is not returned. What is its value?
Solution:
Recall that the value is -1/19 of your bet (apr. -5%, with
00). For best value use $5 of your money. The payoff:
(18/19)*5 (from coupon) - (1/19)*5 (from your $5) =(17/19)*5 ($)
(appr. $4.47).
I used 4 coupons (from 2 fun books in Cains), so expected payoff
was about $17.89. Factual payoff was $20 (Aussi dolars)
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Quiz.
Restricted Nim1. 2, or 5 in one move. 25 stones in one pile. 10
points.
Solution:
0 1 2 3 4 5 6 7 8... L W W
L W W...
The period 3 is underlined. Winning moves:
3n: L (give up against a perfect player).
3n+1 (including 25) : Take 1.
3n+2: Take 2.