So I attempted to analyze this event card to find out the Nash equilibrium that would make everyone relatively content with their choice. I did this mostly for fun and I had time at hand. I went a bit methodic and noted even the most obvious. I did this with my entry-level, limited knowledge of game theory, there are approaches that require deeper skills. For anyone interested, I am going to write my notes below.
According to decision theory, the choice is obviously rock/dynamite with the expected value of 10 as max. But this is a multiplayer game, so it doesn't apply. This problem is under the domain of game theory. This is because if R/D was the absolute choice, then everyone would go for it only to be trashed by a troll team that goes for 4/4 paper. Left side is Player 1, top side is Player 2. Each colliding choices (cells) only apply to two players (elements teams), 6,-6 means Player 1 wins 6 upgrades, Player 2 loses 6 upgrades as a result of this choice.
Analysis of EC with game theory:
- One player's win is other player's loss. Because if one side loses, they will have a disadvantage during the match. Players are not indifferent to other sides win. One side having 6 ups means other has 6 fewer ups
- This is a two player, symmetrical, simultaneous, cycling, zero-sum, non-phased, non-cooperative game. Simultaneous because the choices are revealed at the same time (not by taking turns), cycling because if one side knew the other one's choice, he would change his decision, and this would go on infinitely. Zero-sum because there is no case where both parties win or lose and the sum of win+loss is zero. Non-phased because there is no second step of the game after both choices are revealed (which would require phased-game solutions or Markov chains). Non-cooperative because alliances are forbidden
- Game is not in Nash equilibrium. Because there is no mutual best response strategy. To see that, you put an asterisk to every highest profit on each row and column. If there is a cell with two asterisks, then both players go for that option and you get a mutual best response
- No strictly dominating strategy. Because any single option (pure strategy) for a player is not greater than his other options
- No pure strategy Nash equilibrium. Because no single outcome (cell/intersection) satisfies both players. There is always a more profitable deviation in the same column of that cell for player 1 or row for player 2
- No weakly dominating strategy. Same as above, but deviation doesn't have to be more profitable, equal profit is enough.
- This game has finite number of players and has finite number of options. So there MUST be a Nash equilibrium in which both parties are indifferent.
The solution to Nash equilibrium is a probabilistic distribution over multiple strategies. An example would be; "Team Water selects two strategies, Paper/Dynamite & Scissors/Dynamite. we randomize our choice but with the weighted numbers of Nash equilibrium. Say P/D is 1/10 and S/D is 9/10. We will roll a d10 dice with 9/10 sides saying choose S/D".
At this point, I tried to solve probabilistic distribution between only two strategies to find the equilibrium but there is no 2 strategy combination that can possibly have zero-sum. Below is an example,
The red cells cannot possibly be zero for either player as P1 is always positive (2 & 4) or P2 is always negative (-2 & -4) and neither can be zero when subtracted from each other (after being multiplied with their distribution. also, other cells of the row could be). So you cannot mix those 2 strategies and reach an equilibrium. I tried the combinations but none of the two strategy combinations seem to solve it. If there were any, I could attempt calculating distribution. There is a solution for 3 or 4 or 5 or 6 mixed strategies but the formulas get complicated for me to do by hand. I tried to find some online GT calculator for it and I actually found for 5x5 matrix also an application on MIT website. But right now they are not capable of solving a matrix of this size. But to give you an idea, the equilibrium solution is a weighted dice rolling with some of the strategies are written on more sides. Aside from that, people are not fully rational. So some may act predictable, some can mindgate the others or just be very lucky about their choices. If I can manage find a calculator for 6x6 mixed strategy solution, I am going to post it with a suggestion for everyone. But in any case, modeling is half the solution.
