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The game is not, however, a dominance PD. Indeed, there is no dominant move for either player. It is commonly believed that rational self-interested players will reach a nash equilibrium even when neither player has a dominant move.
To preserve the symmetry between the players that characterizes the ordinary PD, we may wish to modify the asynchronous game. Let us take extended PD to be played in stages.
Next a referee determines who moves first, giving each player an equal chance. Finally the outcome is computed in the appropriate way. It is straightforward, but tedious, to calculate the entire eight by eight payoff matrix. It may be worth noting that an asynchronous version of the Stag Hunt, unlike the PD, presents few issues of interest.
If the first player does his part in the hunt for stag on day one, the second should do her part on day two. If he hunts hare on day one, she should do likewise on day two. The first player, realizing this, should hunt stag on day one.
So rational players should have no difficulty reaching the cooperative outcome in the asynchronous Stag Hunt. Transparency Another way that conditional moves can be introduced into the PD is by assuming that players have the property that David Gauthier has labeled transparency.
A fully transparent player is one whose intentions are completely visible to others. Thus there may be some theoretical interest in investigations of PDs with transparent players.
Such players could presumably execute conditional strategies more sophisticated than those of the non- transparent extended game players, strategies, for example that are conditional on the conditional strategies employed by others.
There is some difficulty, however, in determining exactly what strategies are feasible for such players.
There is no way that both these strategies could be satisfied. Nigel Howard, who was probably the first to study such conditional strategies systematically, avoided this difficulty by insisting on a rigidly typed hierarchy of games. The lesson of all this for rational action is not clear. Suppose two players in a PD were sufficiently transparent to employ the conditional strategies of higher level games.
How do they decide what level game to play? Who chooses the imitation move and who chooses reciprocal cooperation? To make a move in a higher level game is presumably to form an intention observable by the other player.
But why should either player expect the intention to be carried out if there is benefit in ignoring it? Conditional strategies have a more convincing application when we take our inquiry as directed, not towards playing the PD, but as designing agents who would play it well with a variety of likely opponents.
This is the viewpoint of Danielson. Howard for an earlier enlightening discussion of this viewpoint. A conditional strategy is not an intention that a player forms as a move in a game, but a deterministic algorithm defining a kind of player.
An agent is simply a computer program, which can contain lines permitting other programs to read and execute it. To be successful a program should be able to move when paired with a variety of other programs, including copies of itself, and it should be able to get valuable outcomes.
There is some vagueness in the criteria of success. Here, where any two programs can be paired, that approach is senseless. Nevertheless, certain programs seem to do well when paired with a wide variety of players.
One is a version of the strategy that Gauthier has advocated as constrained maximization. It is not clear how a program implementing it would move if indeed it does move when paired with itself. Danielson is able to construct an approximation to constrained maximization, however, that does cooperate with itself.
Nevertheless it does move and score well against familiar strategies. Again, it is not clear that the strategy as formulated above allows it to cooperate or make any move with itself, but Danielson is able to construct an approximation that does.
Finite Iteration Many of the situations that are alleged to have the structure of the PD, like defense appropriations of military rivals or price setting for duopolistic firms are better modeled by an iterated version of the game in which players play the PD repeatedly, retaining access at each round to the results of all previous rounds.Ensure availability and sustainable management of water and sanitation for all.
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