Game Theory Voting Utilities

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Here is my full matrix of scores (on right is the key).

Matrix of choices

You will note player 1 only loses two times out of nine, whereas the other two players lose four times out of nine. The only way they can improve their position is by co-operating. This is negated if the voting is secret.

The Nash equilibrium requirement of being aware of the other players strategies is negated if voting is secret, so there is technically no Nash equilibrium. However, if you do not make the voting secret then player 1 (who is mildly dominant) can be manoeuvred by the other two players who should eliminate their dominated strategies, resulting in player 2 & 3 voting the same, which will force player 1 to also follow suit. Hence if the voting is not secret it is dominance solveable and there are three potential Nash equilibria where all three players vote the same way.

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Updated on July 29, 2020

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  • boutiquebankerecon
    boutiquebankerecon over 3 years
    1. We have three players, $1,2,$ and $3 .$ They must vote between three options, $a, b,$ and c. Their utilities for these three options are:
      \begin{array}{llll} & 1 & 2 & 3 \\ a & 3 & 2 & 1 \\ b & 2 & 1 & 3 \\ c & 1 & 3 & 2 \end{array}
      Note: This is not the normal form $-$ it's just a table for summarizing the utilities. Each agent cares only about the outcome, not what he votes for. So, for example, if $a$ wins, regardless of how it wins, 1 's utility is 3,2 's utility is $2,$ and 3 's utility is $1 .$

    The voting is by secret ballot. Each player votes for one alternative. The outcome is determined by majority rule whenever possible, so if any option gets two or more votes, it wins. If there is a three-way tie, the winner is whatever outcome player 1 voted for.

    Write this as a game in normal form. Find the pure strategy Nash equilibria. Is this game dominance solvable? If so, what is the solution?

    So far, I've managed to come up with this solution:

    !

    But as far as here...I can convert this into payoffs, however I'm unsure of how to figure out the Nash equilibria as when we convert from Player 1->3 it starts to confuse me. Any clear answer will be much appreciated. Thanks!

    • boutiquebankerecon
      boutiquebankerecon about 9 years
      I believe Nash Equilibrium is: a,c,c & there is no strict dominance...anyone care to verify?
  • boutiquebankerecon
    boutiquebankerecon about 9 years
    not sure about that...what are your thoughts on the dominance solvable?