I was joking with Eric Kim, a classmate of mine, this week that the game he was studying could be reduced to a certain position of nim. His work on Richmann games studies another very interesting unsolved game theory problem. A Richmann game is a combinatorial strategy game. Given a finite graph and a token placed on a starting vertex (point) of the graph, the game begins. Each player also starts with a fixed amount of money. Then on each turn, both players secretly write on a card how much of their money they are willing to give in order to move the token closer to their destination vertex. The person who bids more is then allowed to move the token one step closer to their destination vertex. However, the player who made the move must pay their opponent how much ever they had on their card. The game ends once one player gets the token to their destination point.
Lots of interesting theorems come out of this game. For example, there is a ratio, such that if the amount of money one player has divided by the amount of money the other player has is exceeding that value, victory will be guaranteed for one player. The game also allows for Richmann values to be designated to positions and Richmann costs be designated to moves.
Just within the setup of the game, I noted some similarities to nim. In both games, we are able to assign value to certain positions. Both games also follow the strict alternating turns pattern.
As I was noting the similarities, things started to add up for me. I saw (very intuitively and roughly and not very mathematically at all) how the games seemed similar. So I stared at my notebook for some time. However, something bothered me. A simple rule of Richmann games seemed out of place. The rule was that addressing the occurence of a tie. If two players both are willing to wager the same amount of money, the rules of Richmann games say to flip a coin, with the winner of the coin flip allowed to move the token towards their destination. Given that the probability that two players wager the same value is quite literally zero (since they can wager any non-negative real number, of which there are infinite), this shouldn't actually affect the game. This still bothered me, so I started to look into this a bit more (yayy for new problems). Thats where my week has been in terms of nim.
With Differential Geometry, I think I've caught up, but I wont really know till the midterm next week. Hopefully I'll be able to answer all the problems this time.
Thats all for now.
Thanks for reading~
Lots of interesting theorems come out of this game. For example, there is a ratio, such that if the amount of money one player has divided by the amount of money the other player has is exceeding that value, victory will be guaranteed for one player. The game also allows for Richmann values to be designated to positions and Richmann costs be designated to moves.
Just within the setup of the game, I noted some similarities to nim. In both games, we are able to assign value to certain positions. Both games also follow the strict alternating turns pattern.
As I was noting the similarities, things started to add up for me. I saw (very intuitively and roughly and not very mathematically at all) how the games seemed similar. So I stared at my notebook for some time. However, something bothered me. A simple rule of Richmann games seemed out of place. The rule was that addressing the occurence of a tie. If two players both are willing to wager the same amount of money, the rules of Richmann games say to flip a coin, with the winner of the coin flip allowed to move the token towards their destination. Given that the probability that two players wager the same value is quite literally zero (since they can wager any non-negative real number, of which there are infinite), this shouldn't actually affect the game. This still bothered me, so I started to look into this a bit more (yayy for new problems). Thats where my week has been in terms of nim.
With Differential Geometry, I think I've caught up, but I wont really know till the midterm next week. Hopefully I'll be able to answer all the problems this time.
Thats all for now.
Thanks for reading~
Nice to see how you can look at other games outside of yours and compare them, hopefully you find something else interesting! Perhaps it could even aid in your research of your own game~
ReplyDeleteWhy did the coin flip bother you so much? Good luck on your midterm!
ReplyDeleteFascinating! Are there other games like nim and richmann? Are these games designed solely for looking at the mathematics and statistics behind it?
ReplyDeleteDid the coin flip rule seem out of place just because it was unnecessary? I would think it's better to address the possibility of a tie than to not, even if it would never happen. Plus if the players start with a fixed amount of money, and they're required to give their bid to the other player if they bid higher, wouldn't there not be an infinite amount of possible bids?
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