Lecture 1 Flashcards
Define Game Theory
Game theory is the conceptualization of decisionmaking where decisions are shaped by simultaneous actions.
Define finite strategic games.
- Finite number of players.
- For each player, a non empty action set. (which could be different for each player.)
- For each player, preferences represented by utility functions.
Use the Moriarty and Sherlock Holmes as an example. Define the actions, and their respective utilities.
Actions: {Dover, Canterbury}
Players: {Sherlock, Moriarty}
Utility…
Define the concept of strategy.
A strategy is not an action, it is a complete set of actions which the player might take given different context. It can be represented by a probability distribution.
How are individual preferences characterized for each player?
The expected utility over all strategies of all players.
Define a Nash Equilibrium.
A strategy profile s is an NE if
v_j(s*j, s*-j) >= v_j(s_j, s_-j) for all j
with v_j being individual preferences and s_j being individual strategies.
What is the golden rule of mixed equilibrium?
If players mix, they must be indifferent across all outcomes.
If there exists a NE in a Zero Sum Game. What must be true about the individual preferences of each player?
That the min(s_i) max(s_j) {v_j(s_j, s_i)} = max(s_j) min(s_i){v_j(s_j, s_i)}
Give on specificity of optimal strategies in Zero Sum Games.
If a strategy is optimal in a ZSG, then such a strategy is also optimal in ZSG where utility is monotonically transformed.
Define the bargaining problem in the context of cooperative Game Theory.
- A compact set of outcomes X
- Two continuous utility functions.
- An outcome d in the set X called the disagreement outcome. Both players prefer all outcomes to the disagreement outcome.
What is an important feature of ZSG?
One player’s win is the other’s loss. u_1(s) = -u_2(s)
What does the Nash BArgaining Solution outcome maximize?
The product of added utilities w.r.t the status-quo outcome.
Prove the if of Von Neuman Proposition.
Prove the “and only if” of the Von Neuman Propositon.
Prove the if of the following poposition.
Prove the “and only if” of the following proposition.
