Universiteit van Amsterdam
Games and Strategy
Summary
October 2025
, 1 Week 1
Choice rules
• Strictly prefer − > a1 ≻ a2
• Weakly prefer − > a1 ≿ a2
• Indifferent − > a1 ∼ a2
Dominance
• Prefer outcome of a1 to the outcome of a2 in every state − > a1 (s) ≻ a2 (s)
• Strict − > Action a2 is strictly dominated by action a1 − > a1 ≻ a2
• Weak − > Weakly prefer outcome of a1 to outcome of a2 in every state and strictly prefer outcome
of a1 to outcome of a2 in some state
• Limitation − > Doesn’t rank all possible actions
Properties
• Completeness − > For every a1 , a2 we want a1 ≻ a2 , a1 ∼ a2 or a2 ≻ a1
• Transitivity − > a1 ≻ a2 and a2 ≻ a3 then a1 ≻ a3 must hold. Otherwise it creates cycles, which
could lead to a money pump.
Money Pump Example
• p(s1 ) = p(s2 ) = 25%
• p(s1 ors2 ) = 60%, which means you get $60 for sure or $100 if s1 or s2 happens.
• Offer to receive $59 now, but pay $100 if s1 or s2 happens. The other party will accept, which
means you pay $59 but receive $60 from this bet.
• Offer to pay $26 now, but get $100 if s1 / s2 happens. The other party will accept, which means
you pay $26 instead of $25 from this bet.
• $59 - $26 - $26 = $7
• You make a $7 profit. So no matter what state happens, the other party will never get any money.
Maximin
• Maximize worst possible outcome
• Limitation − > Does not take likelihood into account.
Expected Utility
• u(a) = p(s1 )v(a(s1 )) + p(s2 )v(a(s2 )) + ....
• Where p(s1 ) is the probability that state 1 happens.
• v(a(s1 )) is the outcome of action a if state 1 happens.
Inferring Probabilities
• Risk-neutral − > u(x) = x
• Get $1 if s happens or $x for sure − > p(s) ∗ 1 = x
1
Games and Strategy
Summary
October 2025
, 1 Week 1
Choice rules
• Strictly prefer − > a1 ≻ a2
• Weakly prefer − > a1 ≿ a2
• Indifferent − > a1 ∼ a2
Dominance
• Prefer outcome of a1 to the outcome of a2 in every state − > a1 (s) ≻ a2 (s)
• Strict − > Action a2 is strictly dominated by action a1 − > a1 ≻ a2
• Weak − > Weakly prefer outcome of a1 to outcome of a2 in every state and strictly prefer outcome
of a1 to outcome of a2 in some state
• Limitation − > Doesn’t rank all possible actions
Properties
• Completeness − > For every a1 , a2 we want a1 ≻ a2 , a1 ∼ a2 or a2 ≻ a1
• Transitivity − > a1 ≻ a2 and a2 ≻ a3 then a1 ≻ a3 must hold. Otherwise it creates cycles, which
could lead to a money pump.
Money Pump Example
• p(s1 ) = p(s2 ) = 25%
• p(s1 ors2 ) = 60%, which means you get $60 for sure or $100 if s1 or s2 happens.
• Offer to receive $59 now, but pay $100 if s1 or s2 happens. The other party will accept, which
means you pay $59 but receive $60 from this bet.
• Offer to pay $26 now, but get $100 if s1 / s2 happens. The other party will accept, which means
you pay $26 instead of $25 from this bet.
• $59 - $26 - $26 = $7
• You make a $7 profit. So no matter what state happens, the other party will never get any money.
Maximin
• Maximize worst possible outcome
• Limitation − > Does not take likelihood into account.
Expected Utility
• u(a) = p(s1 )v(a(s1 )) + p(s2 )v(a(s2 )) + ....
• Where p(s1 ) is the probability that state 1 happens.
• v(a(s1 )) is the outcome of action a if state 1 happens.
Inferring Probabilities
• Risk-neutral − > u(x) = x
• Get $1 if s happens or $x for sure − > p(s) ∗ 1 = x
1
