Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
Scarica il documento per vederlo tutto.
vuoi
o PayPal
tutte le volte che vuoi
PROFILE THAT GUARANTEES THE SAME OR EVEN GREATER PAYOFF FOR BOTH PLAYER, OR AT LEAST FOR ONE PLAYER WITHOUT THE OTHER PLAYER BEING WORST OFF-BEST RESPONSE
• If more than one undominated strategy, what is good (rational?) choice
• player should choose strategy which maximizes (expected) payoff given own belief about other player’s choice. Such a strategy is called best response, or best reply
• It is best response to own belief
Best response do not exist in some cases
In FINITE games a BR always exists.
A Rational player will never play a strategy that is not a best response as well as a dominated strategy.
GAMES WITH 2 PLAYERS:
If a strategy is strictly dominated it is never a best response
If a strategy is never a best response it is strictly dominated
GAMES WITH 3 OR MORE PLAYER:
If a strategy is strictly dominated it is never a best response
If a strategy is never a best response IT DOES NOT IMPLY that the strategy is strictly dominated
ESERCIZIO: Andiamo a cercare
le strategie che non sono dominated (undominated). È strictly dominated quindi L è undominated Se la strategia è undominted è una best response Procedure for calculating Bi = UDi : 1. Look for strategies that are best responses to the simplest beliefs—those beliefs that put all probability on just one of the other player’s strategies. These best responses are obviously in the set Bi so they are also in UDi . 2. Look for strategies that are dominated by other pure strategies; these dominated strategies are not in UDi and thus they are also not in Bi . 3. Test each remaining strategy to see if it is dominated by a mixed strategy. This final step is the most difficult, but if you are lucky, then you will rarely have to perform it. VEDI QUADERNO PER ESERCIZIO RATIONALIZABILITY AND ITERATED DOMINANCE A player is thought to be rational if (1) through some cognitive process, he forms a belief about the strategies of the others, and (2) given this belief, he• Can we try to achieve further results relying on the assumptions that players are rational?
• If a player believes that his opponent is rational, what probability will he assign to strictly dominated strategies of the opponent?
• Suppose that rationality of players is common knowledge, that is: all players are rational, all players know that all players are rational, all players know that all players know that all players are rational and so on . . .
• Common knowledge of rationality can impose restrictions on a player's beliefs about other players' choices
Player has no strictly dominated strategy and so the set of UD coincides with the set of available strategies (A, B). For player 2 the set of UD = (Y, Z).
IF RATIONALITY IS A COMMON KNOWLEDGE: since player 1 know that payer 2 is rational, he knows that player 2 will
never choose x. for player 1, A is strictly dominated by B. so it will never choose strategy A. Y is strictly dominated forplayee 2 so it will never choose strategy Y. so it remains a unique strategyprofile (B,Z).This procedure is called iterate dominance. Remark:- not playing a dominated strategy only requires individual rationality;
- iterated dominance requires common knowledge of individual rationality, since players eliminate own strategies which are dominated if opponents do not play their dominated strategies;
- common knowledge ensures that we can perform any number of eliminations (infinite hierarchy).
never a best response.
- In particular, we can perform the iterative removal of strategies that are never a best response relying on the assumption that rationality is common knowledge.
- The rationalizable strategies are those that survive the iterated elimination of strategies that are never best responses.
A strategy is rationalizable for a player if it is a best response given the player's belief and if such belief is not in conflict with rationality being common knowledge.
STESSO ESEMPIO
- Player 2, X is strictly dominated and so can never be a best response. Quindi possiamo eliminarla.
- Dato che il player due non ha alter best response, ci concentriamo su player one. A is strictly dominated and so is never a best response e quindi possiamo eliminarla.
- Y is strictly dominated for player 2 so it is not a best response and will never choose strategy Y. so it remains a unique strategy profile (B,Z).
Can X be rationalizable for player 2?
There should be a belief such that X is a BR.
But we know that X is strictly dominated and never a best response so there is no belief (theta) that can support the fact that X is (a best response and so) rationalizable.
Strategy A can be rationalizable for player 1?
A is a best response if player one believes that player 2 will choose X (first player).
There is no belief about what player 2 believes about what player 1 will choose that can lead to adopting X.
So, A cannot be a rationalizable strategy.
Can Y be rationalizable for player 2?
CAN B BE RATIONALIZED FOR PLAYER 1?
CAN Z BE RATIONALIZED FOR PLAYER 2?
STRATEGIC UNCERTAINTY (different FROM COMMON KNOWLEDGE) = a player knows that there are multiple plausible strategy profiles, therefore is uncertain about how the other players will play the game.
IMPORTANT: Rationalizability merely requires that the players' beliefs and behavior be consistent with common knowledge of rationality.
Rationality does not require beliefs to be correct. A player can choose a strategy with the belief that a
- people best respond to their beliefs
- beliefs are not in conflict with rationality being common knowledgeamong players
- Belief and behavior of each player are consistent but they may be notconsistent with the behavior and beliefs of the opponents.
- So, we do not assume player's beliefs are consistent with actual opponents' strategies.
- We will now introduce a equilibrium concepts that also establishes a linkbetween a player's beliefand the actual behavior of his opponents.
- We will denote a strategy profile (a strategy for each player) s as follows:
- No players have a strictly profitable deviation.
- The entries with both payoffs underlined are associated to Nash equilibria of the game: (D, D)
- (D, D) is also the unique strategy profile that survives the iterated removal of strictly dominated strategies
- A Nash equilibrium should not be Pareto efficient, (D, D) is not Pareto inefficient since (C, C) is more efficient than (D, D)
- Nash equilibrium does not eliminate problem of coordination, that is if we require that beliefs are correct in addition to the fact that rationality is common knowledge, coordination on inefficient outcomes can still occur.
- Strategic tension between coordination and social welfare. Pareto efficient allocations can emerge in equilibrium even though there are Pareto efficient equilibrium outcomes.
- This occurs despite all players have the same preferences over
outcomes (unlike battle of the sexes)
- Remember, even though we underline payoffs, Nash equilibria are strategy profiles, not payoffs.
- If you had to report the Nash equilibria of the Pareto Coordination game, you should write (A, A) and (B, B) not the associated payoffs.
- Nash equilibria are immune to unilateral deviations: no player has incentive to deviate from his strategy given that the opponents do not deviate.
- Nash equilibria can be vulnerable to joint deviations by a coalition of players.
Remarks:
- Nash equilibrium requires that beliefs of players are correct;
- each Nash equilibrium is a rationalizable strategy profile;
- there can be more than one Nash equilibrium;
- a Nash equilibrium can be Pareto inefficient;
- some games have no Nash equilibrium (in pure strategies)
DOMANDA: as a corollary of the Nash Theorem, every finite extensive game with perfect information has always at least one subgame perfect equilibrium
When strategies are continuous variables,
We must work with best response functions (provided some conditions are satisfied).
Consider the following normal-form game: S1 = S2 = [0, 1] and payoff functions
PLAYER 1: *faccio la derivate
PLAYER 2: MIXED STRATEGY NASH EQUILIBRIA
A mixed-strategy Nash equilibrium is a mixed-strategy profile having the property that no player could increase his or her payoff by switching to any other strategy, given the other player's strategy.
For a mixed strategy to be a best response (as required in the definition), it must put positive probability only on pure strategies that are best responses. This demonstrates how to calculate a mixed-strategy Nash equilibrium.
The two NE in pure strategies are NL and LN.
In addition to these pure-strategy equilibria, there is also a mixed-strategy equilibrium. To find it, recall what must hold in a mixed-strategy equilibrium: a player must achieve a best response by selecting a mixed strategy.
Let's guess that firm X mixes between L and N. If this strategy
is optimal for firm X (in response to the other firm's strategy), then it must be that the expected payoff from playing L equals the expected payoff from playing N; otherwise, firm X would strictly prefer to pick either L or N.
Stesso discorso per player 2
ORA VAI ALLE MIXED:
Per trovare la mixed best response trovi il punto di intersezione:
Note that every pure-strategy equilibrium can also be considered a mixed-strategy equilibrium—where all probability is put on one pure strategy. All of the games analyzed thus far have at least one equilibrium (in pure or mixed strategies).
Procedure to characterize mix
Accedi a tutti gli appunti
Tutor AI: studia meglio e in meno tempo
