players: making a move making a move all at once

example: prisoner's dilemma

• as prisoners cannot communicate, their action: pseudo simultaneous
• if both remains silent: 2 yr to each
• if both confess: 4 yr to each
• if only one confess: 1 yr to confessor, 5 yr to another

👨‍🏫 consider it as a matrix

assume rational behavior of prisoners:

• working for his / her own interest
  • not about their old colleague anymore
• maximizing their benefit, or minimizing their cost in this case

for total sum: $2\backslash 2$ is ideal

however: do case analysis

• if $P_2$ confess: $P_1$ better confess too, it minimizes the year
• if $P_2$ remains silent: $P_1$ better confess, again
  • it minimizes the year
• thus: $P_1$ is better confess
• ... as the table is symmetric: both gets to confess
  • and gets $4$ year each

• a set $S_i$ for strategies for player $i$
• a payoff function $u_i:S_1\times S_2\times \dots S_n\to \mathbb{R}$

• 👨‍🏫 definition of "rational"!

• the outcome: $s=(s_1,s_2,\dots s_n)$

• $\forall s_{-i}\forall s_i^{\prime }{u}_i(s_i,s_{-1})\ge u_i(s_i^{\prime },s_{-i})$
• no matter what other people do: doing $s_i$ is always the best

• e.g. for $k$ pollutants: every country loses $k$
  • additional 5 for countries who stopped polluting

• thus: every country will have cost of $n$
• while we could have cost of $5$ each otherwise

• 👨‍🎓 so should we be irrational..?

boy & girl: want to spend evening together

• boy: wants to watch baseball
• girl: wants to watch softball

"payoff" matrix

no dominant strategy exists

• we must find: equilibrium
• e.g. if we are in baseball, baseball:
  • no one would want to change: as neighbors provide less benefit
  • same for softball, softball
• 👨‍🏫 aka Nash equilibrium

• $\forall i,\forall s_i^{\prime }\in S_i,u_i(s_i,s_{-i})\ge u_i(s_i^{\prime },s_{-i})$
• i.e. no one has motivation to change

• but psychology is not science

• Rock Paper Scissors!

guessing the opponent's (dishonest) coin flip (=side choice!)

no "pure" Nash equilibrium

• ${\Delta }_i$: set of strategies for player $i$

• thus: redefine rational
• into maximizing "expected profit"

• then outcome: $s=(s_1,s_2,\dots ,s_n)$ where $\forall i,s_i\sim {\delta }_i$

• $\sigma =({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n)\in {\Delta }_1\times {\Delta }_2\times \dots ,\times {\Delta }_n$
• a Nash equilibrium if:
  • $\forall i,\forall {\sigma }_i^{\prime },E[u_i({\sigma }_i,{\sigma }_{-1})]\ge E[u_i({\sigma }_i^{\prime },{\sigma }_{-1})]$

• ${\sigma }_1=(1/3,1/3,1/3)$
• ${\sigma }_2=(1/3,1/3,1/3)$
• 👨‍🏫 you can compute the expected value yourself

• any $n$-player game in which every $S_i$ is finite has a mixed Nash equilibrium
• 👨‍🏫 proof: not coverage of 2711H. However, it's around Ch. 20 in the book
  • w/ infinite players or infinite strategies: it doesn't hold
  • sadly, we don't know how to compute the equilibrium in polynomial time
• 👨‍🏫 just know that this exist!