Nash Equilibrium Flashcards
DEFINITION: Game
- A situation where an outcome arises as the result of interdependent decisions.
- A game is a situation of strategic decision-making –> Strategy depends on decisions by other players.
- Theoretically limitless number of players, usually small number
- Players have different available courses of actions
- Key: Different combinations of choices lead to different outcomes
Assumptions in standard game theory
- Complete Information (Courses of actions, possible outcomes. which action combinations lead to which outcomes. all payoffs associated with the different outcomes)
- Rationality (maximization of utility)
- Common knowledge of rationality (I know, that the other one is rational and the other one knows..)
–> Goal of game theory is to model, not to give you advice
Pareto-inferior outcomes
- Can make both better off by moving to a different (Pareto-superior) outcome
- Can make one player better off and keep the other player’s payoff the same
Pareto-efficiency
No other outcome is Pareto-superior (Impossible, by switching to another outcome, to make one player better off without making the other player worse off)
Strictly Dominant strategy
Yields a strictly greater payoff than all other strategies regardless of the other players’ strategies
Weakly Dominant strategy
Best possible response among others
DEFINTION: Nash Equilibrium
Doing the best I can given what you are doing and vice versa (Mutual best responses)
–> Any equilibrium in dominant strategies is a NE but not vice versa
–> often, games have more than one NE, sometimes they have many
Assumptions Nash Equilibrium
- Rationality and common knowledge of rationality
- Ability to think strategically (.e., to put oneself in the other player’s shoes)
- Belief that other player play’s Nash Equilibrium as well
Nash Equilibrium vs. Normal Equilibrium
NE: I’m doing the best I can given what you are doing and vice versa
Normal Equilibrium: I’m doing the best no matter what you are doing and vice versa
Iterated Elimination of Weakly Dominated Strategies (IEWDS)
Every IEWDS solution is a NE but not vice versa
–> Look if any rows / columns are dominated by any other and cross out step-by-step
Problem: Order of elimination can matter
NE with 3 Players
Player 1 & 2 as usual
Player 3 compare each cell from the first table with the second table to find the better value and underline this
DEFINITON: Common Consequence Effect
- If you have a gamble with safe outcome (p=1), players tend to prefer this one compared to gambles with small chance of getting nothing, even though the latter game has higher EV
- If you have two gambles, where nothing is safe, players tend to choose the one with higher EV
–> If there is a safe alternative, this is preferred (loss is weighted higher than gain)
–> If there is no safe alternative, higher EV is prefered
Having reached a Nash Equilibrium means that…
… no individual player has any reason to switch his or her strategy unilaterally
The guessing game
- bounded rationality prevents players from fully reasoning their way to the equilibrium
- Rational players reckon that other players are boundedly rational and best-respond to the behavior they expect from other players
- Multiple possibiites for anticipation of other players actions
–> significant divergence in results
Level-k Model
- Level 0: “It all looks quite random, so I’ll just guess”
- Level 1: “I think that most people won’t have a clue what to choose and so the average number will be around 50. Two-third of that would be 33.”
- Level 2: “Assuming that everyone thinks the average will be 50 based on random probability, I expect most choices to be located around 33. I am going to choose two-third of that, 22.”
Level 0: Uninformed random play
Level k: Best responding to level (k-1)
NE at level ∞
