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The mathematical theory of game theory is about finding optimal srategies for games such as rock-paper-scissors.
I have heard it claimed several times that game theory was used during World War II by the Americans to decide military strategies but I am unable to find any reliable source, except for some that model the situation afterwards.

Was game theory really used by the HQ or was the situation just left to the gut feeling of the commanders?

I am interested in concrete game theoretic modeling of military situations based on which decision was made, in any war. To be more specific, I mean that a mathematical model was made based on some available options, and the option executed in reality was decided by randomly selecting one of the options, based on the optimal mixed strategy.

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    You are asking about game theory in specific, and not operations research in more general terms?
    – o.m.
    Jul 18 at 4:40
  • Yes, exactly so.
    – domotorp
    Jul 18 at 4:52
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    @rs.29, while war game was called a game and could have a referee, it does not strictly use game theory in the sense of mathematics.
    – o.m.
    Jul 18 at 10:03
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    @rs.29, game theory has a very specific meaning, hence my initial comment. Not every insight drawn from games is a game theory.
    – o.m.
    Jul 18 at 14:12
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    @Italian I don't think game theory dates from 1951, I would rather cite Neumann, 1928-1944, see en.wikipedia.org/wiki/Theory_of_Games_and_Economic_Behavior. Also, in a link included in my original question, see the claim: "game theory has been heavily involved in decision making for battle plans during World War 2".
    – domotorp
    Jul 18 at 18:50
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Game theory was indeed, developed around the time of World War II. The example generally cited is the dilemma facing American Air Force General George Kenney in New Guinea.

There were two east-west routes across New Guinea for Japanese convoys, the cloudy northern route, and the sunny southern route. A convoy used the southern route, one of two things would happen: The convoy would be destroyed if Kenney directed his planes that way, and unscathed if he chose to patrol the northern route.

A convoy using the northern route would be "damaged," but some were likely to survive if attacked. (Of course, they could be "unscathed" if Kenney patrolled the southern route.)

The Japanese opted for the northern route, "some damage, some survival" versus the all-or-nothing southern route. Anticipating this, Kenney patrolled the northern route. And the "saddle point" result of "some damage" was achieved.

A similar decision faced Eisenhower during the invasion of Italy. The two choices were Rome, and Salerno (southern Italy). In the latter location, fighter cover would ensure at least an American beachhead, even against German opposition. If the landing was at Rome, it would either shorten the Italian campaign by a year (capture Rome in 1943 instead of 1944 if unopposed), or be destroyed by German resistance with no fighter protection (a real setback).

Eisenhower opted for the safer choice (sure beachhead establishment, longer slog up the peninsula). Germany's "Smiling Albert" Kesselring anticipated this and pushed German forces to the south, mostly bypassing Rome along the way.

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    These more seem to fall in the category of minimizing worst-case loss rather than using a mixed strategy where the choice is made randomly, as in an optimal play of rock-paper-scissors. Also, I feel that these were choices left to the 'gut feeling of the commander.'
    – domotorp
    Jul 27 at 4:55
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    This answers the question "have game theorists used WWII as a source of examples" not "was game theory used during WWII" Jul 29 at 11:44

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