Repeated games with complete info Flashcards
Repeated games
In general we can think about a repeated game in contrast to the games we have played before, because they have not been repeated. Here we have the same games, but we will repeat them severeal or infinite times, which will change how the players are thinking.
If we took the prisoners dilema. If that is only played one time the NE would be that both players confess (finks). But if we play it several time the players need to think of, if they could get a higher utility of cooperating instead of non-cooperating.
Another examply is an ologopy, where to firms can choose to cooperate and get a good payoff in all following periods by seeting their prices equal. Or one of them could deviate, setting the price lower than the other and by that getting the entire market and probarbly a better payoff for one period.
How could you get to a situation, where we can sustain cooperation?
The right discount factor.
Finitely repeated games
A game where there is a finite amount of repeatings
Infinitely repeated games
A game whitout a finite amount of repeatings. It goes on foerver.
Trigger strategy
Both players will play the cooperative action, but If someone choose the non-cooperative action, that player will pe punish for it in all following periods into internity. So you can either choose to cooperative or not, because there is no turning back.
Ex. Prisioner dilema (firm dilema): First round of repeative games both players play munk (cooperate). Second round the P2 plays fink (defected), and I P1 plays munk, because he trusted P2 t0 play munk. Because of P2 plays fink one time, P2 will now always and forever play fink, as a punishment. You could say P1 triggered P2, apperentaly pretty bad hehe.
OSD strategy
One shoot deviation strategy, like a normal one shot game.
Stage game
The game that is played in each period of a repeated game is reffered to as a stage game.
Is cooperation possible in a stage game with one equilibrium? If not, when so?
In a stage game with only one unique equilibrium it is not possible to cooperate if its reapated a finite amount of times.
If the stage game has multible equilibriums and is again finite it is now possible to cooperate, if the players also care about their future sufficiently much.
Is it possible to have infinitely cooperation.
If the stage game is repeated infinitly cooperation it is possible, if again the players care about their future.
Dynamic games with imperfect information.
When one player reacts afterwards the other player have played his private information.
Example, now the teacher plays first, but we can´t see what he choose. So when we have to play afterwards we have imperfect information.
The dynamic games with imperfect informatio is the same as the static game. We get the same NE´s
Discount factor, and what values can it take?
The discount factor is an important tool in repeated games, where we look at cooperation, because it defines when two agents want to cooperate or not.
We could call it a kind of patience factor. If you are patient, then you have a lot of bargaining power, because you can tire your opponent out.
The discount factor is always between 0 and 1.
If just under 1 then you value e.g. future payoff higher then todays.
If just over 0 then you value todays payoff higher then future.
If it was exact 1, then you would not discount the future, and you will just get the same payoff forever. On the other hand if it was exact 0, then you would place zero value on future payoff with zero, meaning the you would be in a one-shoot game.
Why will people in the prisoner dilemma not cooperate if the game is finite, but they want to if the game is infinite.
It depends on the discount factor. If the they get a high payoff by deviating, and the other option to cooperate is relatively low and need to be discounted every period, then it could be a good idea to deviate in a finite game. If the game was infinite, and you was able to wait (discount factory) then it could be a good idea to cooperate in this game.
If we are in an infinite game, two players choose to cooperate in every periode into infinity. You get a payoff today on 3. Are you payoff tomorrow and every other period also 3.
No, the value would be discounted with some discount factor, due to fx the alternative cost of not investing the money, maybe cooperation could break down and so on.
Tit for Tat game
In contrast to the trigger strategy, the tit for tat strategy gives several chances to the other player. Basicly its just copying its apponent, which means that it can never win. It can either tie or loos.
There is another strategy, called forgivning tit for tat which requieres two defections before copying, therefore we will not get caught in an echo chamber. But sometime it becomes to forgiving.
E.g. if P2 chose to deviate then P1 playing tit for tat would also deviate in the next round, but if P2 then chosed to go back to cooperating, P1 would give P2 a second chance an play cooperat in the following round.
Is Tit for Tat SPNE.
No, it is not, if the players have different discount factors.
Knife edge case if they have.
