overview

At the start of the lecture, we introduce the “formal ingredients” of a game: the players, their strategies and their payoffs. Then we return to the main lessons from last time: not playing a dominated strategy; and putting ourselves into others’ shoes. We apply these first to defending the Roman Empire against Hannibal; and then to picking a number in the game from last time. We learn that, when you put yourself in someone else’s shoes, you should consider not only their goals, but also how sophisticated are they (are they rational?), and how much do they know about you (do they know that you are rational?). We introduce a new idea: the iterative deletion of dominated strategies. Finally, we discuss the difference between something being known and it being commonly known.

The ingredients of a game

Ingredients of a game are players,strategies, payoffs.

Ingredients Notations
players i,ji,j
a particular strategy of player i sis_i
the set of possible strategies of player i si′s_i^{'}
a strategy profile/vector/list s(all strategies for a particular game)
payoffs of player i Ui(s1,s2,…,sN)U_i(s_1,s_2,\dots,s_N) or Ui(s)U_i(s)

Except notations in above table, now we use “s−is_{-i}” meaning a strategy choice for everybody except player i.
Using these notation, we can give the definition of strict dominance:

Def: player i’s strategy si′s_i^{'} is strictly dominated by player i’s sis_i if:

Ui(si,s−i)>Ui(si′,s−i)foralls−iU_i(s_i,s_{-i})>U_i(s_i^{'},s_{-i}) \hspace{2cm}for \hspace{0.2cm}all\hspace{0.2cm} s_{-i}

Similarly, we can give the definition of weak dominance：

Def: player i’s strategy si′s_i^{'} is weakly dominated by player i’s sis_i if:

Ui(si,s−i)≥Ui(si′,s−i)foralls−iU_i(s_i,s_{-i})\ge U_i(s_i^{'},s_{-i}) \hspace{1cm}for \hspace{0.2cm}all\hspace{0.2cm} s_{-i}

and in addition player i’s payoff from choosing sis_i against s−is_{-i} is strictly better than payoff from choosing si′s_i^{'} against s−is_{-i} for at least one thing that everyone else could do:

Ui(si,s−i>Ui(si′,s−i)forsomes−iU_i(s_i,s_{-i}>U_i(s_i^{'},s_{-i})\hspace{1cm}for\hspace{0.2cm} some \hspace{0.2cm}s_{-i}

Put yourself into other’s shoes and figure out that they are not going to play strongly or weakly dominated strategies. This seems a pretty good way to predict other’s strategy.

A little more about ‘in others’ shoes’

The iterative deletion of dominated strategies

Deleting strictly dominated strategies may convert strictly dominated strategies to weakly dominated strategies for a player who is rational and thinks that his opponent is also rational

Thinking a game in which two player a and b have same trategies, we can reach a conclusion that player a is rational if player a doesn’t choose strongly dominated strategies. And we would say that player a think player b is rational if player a think that player b doesn’t play his strictly dominated strategies. Think more deeply, player a may have new strictly dominated strategies(weakly dominated strategies in the original game), and player a won’t choose new strictly dominated strategies if player a know that player b think that player a is rational.This is an 'in shoes in shoes’argument
In the end, Professor helped us distinguish the difference between common knowledge and mutual knowledge .

Mutual knowledge doesn’t imply common knowledge.
Mutual knowledge: something being known, all people in a group know about an event
Common knowledge: something being commonly known,all people in a group know about an event, they all know that they know the event, they all know that they all know that they know the event, and so on ad infinitum.

I know that my opponent is rational, and I know my opponent know that I am rational···It’s common knowledge.