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We can divide these 26 strategies into four categories: i sink-state C ssC strategies, ii sink-state D ssD strategies, iii suspicious dynamic strategies, and iv hopeful dynamic strategies. Sink-state strategies always-cooperate or always-defect either from the beginning or after some condition is met.
There are eight sink-state strategies in total. Suspicious dynamic strategies start with defection and then move between their defective and cooperative state depending on the other player's decision. Hopeful dynamic strategies do the same, but start with cooperation. There are nine strategies in each of these two categories. For each suspicious dynamic strategy there is a hopeful counterpart. Some of the dynamic strategies do little to optimize their score. For example, Alternator switches between cooperation and defection on each move. These strategies have the design element to stay in state if the opponent has cooperated in the last round but move to state if the opponent has defected; we call this element the conditional cooperation element see Fig.
S1 in File S1. In state , TFT then requires the opponent to cooperate again in order to move back to the cooperative state. WSLS in contrast requires the opponent to defect in order to move back to the cooperative state. But Forgiver moves back to the cooperative state irrespective of the opponents move Fig.
In a game between two TFT players, if by mistake one of them starts to defect, they will continue to defect until another mistake happens. The same is true for WSLS in the alternating game.
Thus WSLS, which is known to be a strong strategy in the simultaneous game, is not expected to do well in the alternating game. Forgiver, on the other hand, is error correcting in the alternating game. It recovers from an accidental defection in three rounds Fig. An asterisk after a move indicates that this move was caused by an error. When the conditional cooperators are playing against a copy of themselves, Forgiver performs very well as it can recover from an accidental defection within three rounds.
Only another mistake can enable them to return to cooperative behavior. When Grim plays against itself and a single defection occurs, it moves to the defection state with no possibility of returning to cooperation.
A stochastic variant of Forgiver is already described in . In this study, strategies are defined by a quadruple where denotes the probability to cooperate after each of the four outcomes , , , and. This stochastic strategy set is studied in the setting of the infinitely-repeated alternating game.
The initial move is irrelevant. In  a strategy close to is victorious in computer simulations of the strictly alternating Prisoner's Dilemma.see
Comando Underground Pub l Duo Alternation
For further discussions see also pp. We calculate the payoff for all pairwise encounters in games of moves of both strategies, thereby obtaining a payoff matrix.
We average over which strategy goes first. Without loss of generality we set. At first we study the case with error rate and an average game length of. Table 1 shows a part of the calculated payoff matrix for six relevant strategies. In the following, we study evolutionary game dynamics  —  with the replicator equation.
Related Alternation 2666
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