I'm reading a bit about Graphical Game Theory, but do not really understand the utility function the model uses.
I'm going to consider a game like matching pennies, but with 3 players: Alice, Bob, and Charlie.
If Alice, Bob, and Charlie all pick heads, (HHH), Alice will lose 1, Bob will gain 1, and Charlie will lose 1 (-1 1 -1).
If Alice and Bob pick heads, and Charlie picks tails (HHT), Alice and Bob will lose 1, and Charlie will gain 1 (-1 -1 1).
Similarly,
$$ \begin{matrix} HHH: \, (-1, \: 1, \: -1) \\ HHT: \, (-1, \: -1, \: 1) \\ HTH: \, (-1, \: 1, \: -1) \\ HTT: \, (1, \: -1, \: -1) \\ THH: \, (1, \: -1, \: -1) \\ THT: \, (-1, \: 1, \: -1) \\ TTH: \, (-1, \: -1, \: 1) \\ TTT: \, (-1, \: 1, \: -1) \end{matrix} $$
How would we get the utility functions from this payoff matrix? Similarly, how would we convert the utility functions back to the payoff matrix?
Thanks for any help.
This was a case of differing terminology; what I knew as a payoff function was called a utility function.
The problem becomes simple after that,
The payoff of Alice is:
$u(A,B,C) = \begin{cases}1 \; ; \; \{A, B, C\} \in \{HHH, HHT, HTH, THT, TTH, TTT\} \\-1 \; ; \; \{A, B, C\} \in \{HTT, THH\}\end{cases}$
The payoffs for others can be derived similarly.