Static and dynamic Games with complete information Flashcards
What is a NE
u_i(si , s*(-1)) \equiv u_i(si , s*(-1))
No player have incentive to deviate.
All players are playing a BR to the other players´ strategies. Then the equilibrium is Nash.
In a finite game, there always exits at least one NE in either mixed or pure strategies.
NE does not require players to make BR at decision nodes that are not reached. So we can have a NE OFF the EP, where the players behavior are not required to be Nash, which means they do not need to play their BR. Therefore a NE can rely on a non-credible threat, where a player is not playing his BR to influence the other player.
What is a pure NE, and when is it unique?
Pure because both players make a BR to the other players´ strategy. Individual deviation is not profitable.
Unique when it is the only NE, given both players´ BR.
What is a mixed NE
If no equilibrium exists in pure strategies, one must exist in mixed strategies.
Then if both players are indifferent in the same spot, then it’s an NE, because none of them have incentive to deviate.
A mixed strategy is a probability distribution over two or more pure strategies.
- That is the players choose randomly among their options in equilibrium.
- If mixtures, are mutual BR the set of strategies is a mixed strategy NE.
example with flip of coin against a mindreader. They end up flipping the coin randomly, giving 50/50 procent chances for each of the four outcomes, se figure 3. Therefore his BR are mutual with the mindreader´s and then the set of strategies are a mixed NE. None of them have incentive to deviate, because they both are indifferent.
What is a proper subgame
One that starts at a singleton informationset, where all information is present.
It is not a proper subgame if there is a cut of information in the singleton iformationset.
Subgame Perfect Nash Equilibrium (SPNE)
Starts at singleton informationset
Players´ strategies constitute a NE in ALL subgames, both ON + OFF the equilibrium path (EP)
Therefore there would only be a unique SPNE, which is the NE, there is ON the EP. The SPNE requires that the players behavior is Nash both ON and OFF EP, so it is not possible for the other player to play a non-credible threat.
It is only natural to impose subgame-perfection if we assume that the players in economic life can se through threats that are not credible.
Explain how the set of pure-strategy Nash equilibria differs from the set of SPNE.
Discuss informally the difference between the definitions of a Nash equilibrium and of a SPNE.
Interpret that difference in economic terms.
In the given example there were 3 pure NE and only 1 SPNE. Therefore the set of pure-strategy NE is larger. It consist of three equilibria, whereas there is one unique SPNE.
The difference between the two equilibrium concepts is that SPNE requires the players´ behavior to be Nash also OFF the EP
Interpret: As a NE can consist of behavior that is not Nash OFF the EP, it can rely on “non-credible threats”.
When we look at figure 1 (in sommer juli 2022), player 2 can threaten to play c if player 1 plays B. If 1 believes this threat (despite the fact that c is not a BR to B), she is now better of playing A (in this example, this is true regardless of whether 2 is expected to make a BR to A). After 1 has played A, 2 optimally chooses b. This outcome yields the payoff 3 to player 2, which is better than what she would get if player 1 chose B. This is why making the threat is useful, provided that player 1 believes it. However, the threat is not credible, in the sense that player 2 would not have an incentive to carry it out if B were chosen. The reason why (A, bc) can be a NE is that the node where 2 choses between c and d is never reached when the equilibrium is played, and a NE does not require players to make BR at decision nodes that are not reached.
It is only natural to impose subgame-perfection if we assume that the players in economic life can se through threats that are not credible.
Credible and non-credible threats, using a 2-player example?
What if player 1 does not know if player 2 is telling the truth?
A threat is used when a player wants to influence the other player´s choice.
Imagine a game with 2 players.
The threat is credible if player 2´s payoff is higher if he deviates in the given subgame. Non-credible if it is not higher. Player 2 wants to make threat, because she could be better of in another NE OFF the EP.
If player 2´s threat is credible, player 1 understand that it will result in undesirable consequences if they due not comply, vice versa.
But it is not given that player 1 knows if player 2 is telling the truth. In that case player 1 has to guess if he thinks she is telling the truth. Or we could be in the opposite case, where player 1 knows if the threat is credible or not, and only then is it natural to impose subgame-perfection.
Subgame
Those we cut of with imperfect information.
Begins at decision node n that is a singleton iformation set. Not the first node of the game, by definition.
Includes all the dicision and terminal nodes.
Does not cut any information sets.
Which Equilibrium do you find the most reasonable/likely, see figur 2
See Figur 2
Aa has the highest reward, and the highest risk.
Bb has zero reward, but lowest risk.
Cc has low reward and high risk.
Therefore we say, that Cc is not the most reasonable. To say which of the last two is the most reasonable we need to know wether how risk avers the agent is.
What is backwards induction?
Backwards induction is a way to solve a game tree, which will give us a NE in all subgames ON and OFF the EP, and a unique SPNE.
Strictly dominated strategy
A strategy is strictly dominated if choosing it yields a strictly lower payoff than some other strategy, regardsless of what strategies the other players choose.
IESDS and a how to
A way of solving a finite normal form game.
How to
1. check if some player has a strictly dominated strategy
2. if so, delete this strategy
3. in the new reduced game, check if some player has a strategy that is strictly dominated.
4. if so, delete this strategy.
5. Repeat until there are no more strictly dominated strategies.
What is the different between a static and dynamic game?
In a dynamic game you can influence during the game, because the players make sequential choices, where in a static game players act simultaneously, without knowledge of the actions taken by other players. So in dynamic games, players have information about the actions taken by previous players before making their own decisions.
example a game of chess. Here the players make sequential moves, where a move from the other player significantly can affect the other players action.
In summary, the key difference between static games and dynamic games is the timing of players’ actions and the availability of information.
Discuss whether we should expect player 1 to obtain a higher equilibrium payoff in the sequential game than in the simultaneous-move game.
In general, it follows that a player cannot be worse off from the opportunity to be the one who chooses first in a sequential game, relative to playing the same game but with simultaneous moves.
Both in the sequential game and the simultaneous-move game, player 2 chooses her action according to the best-response function. The difference is that in the sequential game, player 1 can effectively choose which of the action that maximizes his utility. If he chooses something else then the simultaneous-move game NE´s then that will have a higher payoff (or at least equal).
What is the extensive form game, and how is it different than a normal form game.
The extensive form game is called a game tree or sequential game. The game tree includes information about the order of moves, potential branches based on players’ choices, and the associated payoffs at different points in the game.
The extensive form game allows the players to play sequentially, where they in a normal form game played simultaneously.
