Part 1 Nocooperative Equilibria
in Normal Form Games
Part1-4
Uncertainty, Risk, and Mixed Strategy
NE
prolog
• Incomplete information （不完全訊息）
自然

complete but imperfect information（不完美訊息）
• 虛擬參賽者：自然
以「類型」帶機率的方式出現，不在意報酬。
•
不確定性可能來自「自然」，或來自於人類「有意的」選擇。 著眼
information asymmetry（strategy unpredictability ），提高議價籌碼
• 最適反應：既定對手機率分配下，最大期望報酬之機率分配
不存在純策略，也一定存在混合策略
Mixed-strategy Nash equilibrium
A mixed-strategy Nash equilibrium is mixedstrategy profile having the property that no
player could increase his payoff by switching to
any other strategy, given the other player’s
strategy.
• Note
1.  is compared against each pure strategy
s rather than against all mixed strategies.
2. rationality of mixed strategies
i
'
i
My opponent knows that he cannot out-think me.
s
The pure strategy is not his best strategy. He will
randomize in order to prevent me from out-think
him by choosing probabilities that make the
expected payoffs of my strategies equal. So I may as
well randomize anyway.
'
i
• We look for a mixed strategy for one player
that makes the other player indifferent
between his pure strategies.
•
Graphics for Mixed Strategy
棒球中打者的期望值報酬
pure strategy and mixed strategy NE
• Result：every finite game（having a finite
number of players and a finite strategy space）
has at least one Nash equilibrium（in pure or
mixed strategies）
Equilibriua with Mixed and Pure
Strategies
• Pure strategy NE：（前進, 等候）、（等候，前
進）
• mixed strategy NE：（1/2, 1/2），（1/2, 1/2）
A之預期報酬
• In the coordination game, mixed strategies NE
(（1/2, 1/2），（1/2, 1/2）) is unstable