Threat Power in Sequential Games

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Abstract
The analysis of 2 × 2 ordinal games, in which both players can sequentially move and countermove after an initial outcome is chosen, is extended to repeated play of these games in which one player has ‘threat power’. This power enables this player to threaten the other player with a mutually disadvantageous outcome in order to deter certain moves in the future play of the game. Except for no-conflict games with a mutually best outcome, there are relatively few games in which neither player can threaten the other. Of the games in which one or both players has a threat strategy, those in which one player does enables him to implement an outcome at least as high-ranking for himself as for the other player. In the bulk of games in which both players have threat strategies, threat power is effective—it is always better for a player to have it than for the other player to have it. Where ‘deterrent’ and ‘compellent’ threat power outcomes conflict, deterrent threat power induces a better outcome; conditions for the existence of both kinds of threats in 2 × 2 games, and for the threatener to implement his best outcome in general two-person games, are given. To illustrate the threat-power model, it is applied to the Polish strategic situation in 1980–81. Finally, threat power is compared with other kinds of power that have been proposed in nonrepeated play, and, ominously, ‘Chicken’ is the one game in which threat power is uniquely effective in undermining a nonmyopically stable ‘cooperative’ outcome.

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