Game Theory 101 (#60): Tit-for-Tat Isn't Subgame Perfect

gametheory101.com/courses/game-theory-101/
In this lecture, we see that the punishment strategy tit-for-tat attempts to employ in the repeated prisoner's dilemma is not actually credible. Indeed, if cooperation is so attractive to be worthwhile in the first place, then a player would not want to defect following an opponent's defection. As a result, tit-for-tat is only a subgame perfect equilibrium for a single value of the discount factor, which is an unrealistic knife-edge condition.

[@judy3463]

Thank you so much! You are my lifesaver. Lots of love from Korea :)

[@leylamirzaei825]

are you at the hankuk university maybe ? :)

[@treeman258]

Keep the videos coming. I wonder what would happen in those competitions if some randomness was incorporated into the move each player made? for instance if my bot chooses cooperation, there is a chance that he will actually defect (simulating accidental failures or the like) what would be the best strategy then?

[@Gametheory101]

+treeman258 It's been a while since I have read that literature, but I am pretty sure that tit-for-tat does very well in those cases.

[@treeman258]

+William Spaniel no, if tit for tat was used then whenever a bot "accidentally" defected the two players would go into the 4 1 4 alternation you described.

[@Gametheory101]

+treeman258 By "very well" I meant in comparison to other strategies. You are right that an accident results in the 4 1 4 1 alternation, but: 1) That's still better than what would happen with a grim trigger strategy. 2) Another accident can bring them back to mutual cooperation. 3) Strategies that never cooperate do extremely poorly. Combine those factors together, and you still circumstances that would make tit-for-tat fare comparatively well.

[@sayaks12]

William Spaniel what about forgiving tit-for-tat, that is tit-for-tat which requires the opponent to defect twice in order to defect?

[@jonahprinceton8132]

Instablaster.

[@sammyz1128]

could you make a video about whether grim-trigger is a SPE or not?

[@menscott2581]

It is. The reason is explained in grim-trigger video.

[@BLINKBOXHD]

@@menscott2581Do you mean by identifying which threats are credible? and thus picking the sub game perfect equilibrium through this?

[@Brodysseus113]

I'm looking for a video series that had a simulator game that taught you about this stuff but also allowed for mistakes to be made, and allowed for forgiveness

[@BS33875]

is it that if I don't believe the opponent could want to cooperate I should not cooperate in the beginning?

[@PunmasterSTP]

Tit-for-tat? More like 'Get my hat", because it's raining knowledge!

[@rosokorai9755]

What if there's a 3rd party in the game, an impartial observer - enforcer who would make sure that both parties follow the rules automatically and every single time? Surely then there would not be a subgame inequilibrium. In the real world, there can be all kinds of reasons why a party could not comply EVERY time with the grim trigger... and so it's realistically not very executable. I suppose that's why intermediaries exist in the real world, that of course is costly for both parties both in terms of time and money, but we can also imagine a world in which costly intermediaries are replaced by blockchain enabled smart contracts or trustless protocols.... making the tit for tat strategy work in the long term with no deviation from the happy path. Thoughts?

[@Stop1601]

Question: Wouldn’t the process of introducing a contract system to solve for tit for tat’s lack of credible threat, just result in a grim trigger strategy regardless?

[@user-rn5wd9hl8x]

@5:57 why is it 3+3d >= 4+d ? Isn't it 3(1-d) >= 4+ d? So d =< -1/2 ? (what I mean by d = discount factor)

[@Gametheory101]

From the second line to the third line, multiply everything by (1-d). The left hand side becomes just 3. The right hand side becomes 4/(1+d) + d /(1+d). The rest is easy. The confusing thing here is that (1-d)/(1-d^2) = 1+d.

[@axxenmardok]

This is the first time I didn't get your explanation in this series. Isn't the one-shot deviation principle satisfied if no singular player has a profitable deviation at any one subgame? In the previous example, we looked at one player making a deviation from tit-for-tat, and showed that if delta>1/2, they wouldn't have a profitable deviation. In this example, BOTH players have made a deviation from tit-for-tat, albeit in separate subgames. Doesn't that mean that this is a different strategy profile altogether from the previous video?

[@zacharytaylor2423]

In the delta < 1/2, the profitable deviation is on the path of play. In the delta > 1/2 case, the profitable deviation is off the path of play. "Profitable" means profitable within the relevant subgame, not just profitable on the path of play. Recall that the definition of SPNE requires that if you "start" play from any subgame, the strategies applied to that subgame also correspond to an equilibrium. So for the same reason any deviation on the path of play is enough to conclude the strategy profile is not an equilibrium, any deviation off the path of play is enough to conclude the strategy profile is not subgame perfect.

[@BS33875]

why is it no equilibrium less or more than one half?

[@i-kahn]

Why not consider the first C,D period when the opponent defects in the payoff? I.e. why not 1 + 4ẟ +ẟ^2+4ẟ^3.... EDIT: Nvm got it! Thanks

[@helvinharry569]

Ibrahim Khan can you tell why

[@Brodysseus113]

Look up "The Evolution of Trust"

[@samkim6933]

So isnt tft a SPE when delta is equal to 1/2? Why did you title the video NOT SPE?

[@BS33875]

why will Tit For Tat not work?