[maximin] geometric climbing

A puzzle from The Riddler this week returning to the ranking of climbing competitors in Tokyo. And asking for the maximin score, that is, the worst possible absolute score guaranteeing victory. In the case of eight competitors, a random search for a maximin over 10⁶ draws leads to a value of 48=1x7x8, for a distribution of ranks as follows

[1,]    1    8    8
[2,]    2    6    4
[3,]    3    4    5
[4,]    4    2    6
[5,]    5    5    2
[6,]    6    3    3
[7,]    7    7    1
[8,]    8    1    7

while over seven competitors (the case with men this year, since one of the brothers Mawem got hurt during the qualification), the value is 35=1x5x7, for a distribution of ranks as follows

[1,]    1    7    5
[2,]    2    3    6
[3,]    3    4    3
[4,]    4    5    2
[5,]    5    2    4
[6,]    6    1    7
[7,]    7    6    1

exhibiting a tie in the later case (and no first position for the winners!).

value of a chess game

In our (internal) webinar at CEREMADE today, Miguel Oliu Barton gave a talk on the recent result his student Luc Attia and himself obtained, namely a tractable way of finding the value of a game (when minimax equals maximin), result that got recently published in PNAS:

“Stochastic games were introduced by the Nobel Memorial Prize winner Lloyd Shapley in 1953 to model dynamic interactions in which the environment changes in response to the players’ behavior. The theory of stochastic games and its applications have been studied in several scientific disciplines, including economics, operations research, evolutionary biology, and computer science. In addition, mathematical tools that were used and developed in the study of stochastic games are used by mathematicians and computer scientists in other fields. This paper contributes to the theory of stochastic games by providing a tractable formula for the value of finite competitive stochastic games. This result settles a major open problem which remained unsolved for nearly 40 years.”

While I did not see a direct consequence of this result in regular statistics, I found most interesting the comment made at one point that chess (with forced nullity after repetitions) had a value, by virtue of Zermelo’s theorem. As I had never considered the question (contrary to Shannon!). This value remains unknown.