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We show it by a backward induction.
(It is worth noting that a backward induction, as in [6], cannot be used in our infinite horizon model).
Using a backward induction argument, we analyze the election game in two steps: (1) first, we study the voting game.
If there is even a remote possibility that a player may make a mistake that her 'hand may tremble' then no contradiction is introduced by a player's using a backward induction argument that requires the hypothetical assumption that another player has taken a path that an economically rational player could not choose.
Both players use backward induction to solve the game; backward induction requires that Player I know that Player II knows that Player I is economically rational; but Player II can solve the game only by using a backward induction argument that takes as a premise the failure of Player I to behave in accordance with economic rationality.
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The corresponding online problem is formulated, and a backward induction-based optimal solution is proposed.
The uniqueness of the payoffs at each outcome is only needed to ensure that there is a unique backward induction solution.
The "troika" institutions (the European commission, the European Central Bank, and the International Monetary Fund) have, over the years, relied on a process of backward induction: they set a date (say, the year 2020) and a target for the ratio of nominal debt to national income (say, 120%) that must be achieved before money markets are deemed ready to lend to Greece at reasonable rates.
The "troika" institutions (the European Commission, the European Central Bank, and the International Monetary Fund) have, over the years, relied on a process of backward induction: They set a date (say, the year 2020) and a target for the ratio of nominal debt to national income (say, 120 percent) that must be achieved before money markets are deemed ready to lend to Greece at reasonable rates.
In the language of game theory, it's all a matter of backward induction.
The first is a characterization of iterated removal of strictly dominated strategies (henceforth ISDS), and the second is a characterization of backward induction.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com