Editing Tit for tat (section)

== Problems ==
While Axelrod has empirically shown that the strategy is optimal in some cases of direct competition, two agents playing tit for tat remain vulnerable.  A one-time, single-bit error in either player's interpretation of events can lead to an unending "death spiral": if one agent defects and the opponent cooperates, then both agents will end up alternating cooperate and defect, yielding a lower payoff than if both agents were to continually cooperate. This situation frequently arises in real world conflicts, ranging from schoolyard fights to civil and regional wars. The reason for these issues is that tit for tat is not a [[subgame perfect equilibrium]], except under knife-edge conditions on the [[Discounted utility|discount rate]].<ref>{{cite book
  | last = Gintis
  | first = Herbert
  | title = Game Theory Evolving
  | publisher = [[Princeton University Press]]
  | year = 2000
  | isbn = 978-0-691-00943-8}}</ref>
While this sub-game is not directly reachable by two agents playing tit-for-tat strategies, a strategy must be a [[Nash equilibrium]] in all sub-games to be sub-game perfect.  Further, this sub-game may be reached if any noise is allowed in the agents' signaling.  A sub-game perfect variant of tit for tat known as "contrite tit for tat" may be created by employing a basic reputation mechanism.<ref>{{cite journal |last= Boyd |first= Robert |title= Mistakes Allow Evolutionary Stability in the Repeated Prisoner's Dilemma Game |journal = Journal of Theoretical Biology |volume = 136 |pages = 47–56 |year = 1989 |doi= 10.1016/S0022-5193(89)80188-2 |pmid= 2779259 |issue= 1|bibcode= 1989JThBi.136...47B |citeseerx= 10.1.1.405.507 }}</ref>

Knife-edge is "equilibrium that exists only for exact values of the exogenous variables. If you vary the variables in even the slightest way, knife-edge equilibrium disappear."<ref>{{Cite web|url=http://gametheory101.com/courses/game-theory-101/knife-edge-equilibria/|title=Knife-Edge Equilibria – Game Theory 101|language=en-US|access-date=2018-12-10}}</ref>

Can be both Nash equilibrium and knife-edge equilibrium. Known as knife-edge equilibrium because the equilibrium "rests precariously on" the exact value.

Example:
{| class="wikitable"
|
! Left
! Right
|-
 !Up
|(X, X)
|(0, 0)
|-
! Down
| (0, 0)
| (−X, −X)
|}
Suppose X = 0. There is no profitable deviation from (Down, Left) or from (Up, Right). However, if the value of X deviates by any amount, no matter how small, then the equilibrium no longer stands. It becomes profitable to deviate to up, for example, if X has a value of 0.000001 instead of 0. Thus, the equilibrium is very precarious. In its usage in the Wikipedia article, knife-edge conditions is referring to the fact that very rarely, only when a specific condition is met and, for instance, X, equals a specific value is there an equilibrium.

Tit for two tats could be used to mitigate this problem; see the description below.<ref>{{cite book
  | last = Dawkins
  | first = Richard
  | title = The Selfish Gene
  | publisher = [[Oxford University Press]]
  | year = 1989
  | isbn = 978-0-19-929115-1}}</ref> "Tit for tat with forgiveness" is a similar attempt to escape the death spiral. When the opponent defects, a player employing this strategy will occasionally cooperate on the next move anyway. The exact probability that a player will respond with cooperation depends on the line-up of opponents.

Furthermore, the tit-for-tat strategy is not proved optimal in situations short of total competition.  For example, when the parties are friends it may be best for the friendship when a player cooperates at every step despite occasional deviations by the other player.  Most situations in the real world are less competitive than the total competition in which the tit-for-tat strategy won its competition.

Tit for tat is very different from [[grim trigger]], in that it is forgiving in nature, as it immediately produces cooperation, should the competitor choose to cooperate. Grim trigger on the other hand is the most unforgiving strategy, in the sense even a single defect would the make the player playing using grim trigger defect for the remainder of the game.<ref>{{Cite journal|last=Axelrod|first=Robert|date=2000-01-01|title=On Six Advances in Cooperation Theory|journal=Analyse & Kritik|volume=22|issue=1|pages=130–151 |doi=10.1515/auk-2000-0107|issn=2365-9858|citeseerx=10.1.1.5.6149|s2cid=17399009 }}</ref>