2.5. Investigating dominated strategies and actions
Selecting Tools->Dominance toggles the appearance of a toolbar which can be used to investigate the structure of dominated strategies and actions.
2.5.1. Dominated actions in extensive game
In extensive games, the dominance toolbar controls the elimination of actions which are conditionally dominated. Figure 2-12 illustrates the appearance of the poker game example with the dominance toolbar shown.
Actions may be eliminated based on two criteria:
Strict dominance: the action is always worse than another, regardless of beliefs at the information set;
Strict or weak dominance: there is another action at the information set that is always at least as good as the action, and strictly better in some cases.
removes the dominated action from the game display, as in
Figure 2-13.
The tree layout remains unchanged, including nodes which can only be reached using actions which have been eliminated. To compress the tree to remove the unreachable nodes, check the box labeled Show only reachable nodes. The tree is redisplayed as in Figure 2-14.
For this game, no further actions can be eliminated. In general,
further steps of elimination can be done by again clicking the
next level icon.
.
The toolbar keeps track of the number of levels of elimination currently
shown; the previous level icon
moves up one level of elimination.
The elimination of multiple levels can be automated using the
fast forward icon
,
which iteratively eliminates dominated actions until no further
actions can be eliminated. The rewind icon
restores the display to the full game.
2.5.2. Dominated strategies in strategic games
The dominance toolbar operates in strategic games in the same way as the in the extensive game. Strategies can be eliminated iteratively based on whether they are strictly or weakly dominated.
When the dominance toolbar is shown, the strategic game table contains indicators of strategies that are dominated. For example, Figure 2-15 shows the prisoner's dilemma example with the dominance toolbar shown.
In the prisoner's dilemma, the Cooperate strategy is strictly dominated for both players. This strict dominance is indicated by the solid "X" drawn across the corresponding strategy labels for both players. In addition, the payoffs corresponding to the dominated strategies are also drawn with a solid "X" across them. Thus, any contingency in the table containing at least one "X" is a contingency that can only be reached by at least one player playing a strategy that is dominated.
Strategies that are weakly dominated are similarly indicated, except the "X" shape is drawn using a thinner, dashed line instead of the thick, solid line.
Clicking the next level icon
removes the strictly dominated strategies from the display,
and presents the prisoner's dilemma as in Figure 2-16.
