A useful reduction (SAT -> game)

1
A useful reduction (SAT -gt game)

• Theorem. SAT-solutions correspond to
  mixed-strategy equilibria of the following game
  (each agent randomizes uniformly on support)

SAT Formula
(x1 or -x2) and (-x1 or x2 )
Solutions
x1true, x2true
x1false,x2false
Game
x1
x2
x1
-x1
x2
-x2
(x1 or -x2)
(-x1 or x2)
default
x1
-?,-?
0,-?
- ?,1
-?,-?
-?,-?
-?,-?
2,-?
2,-?
0,-?
x2
-?,-?
-?,-?
-?,-?
- ?,1
2,-?
2,-?
0,-?
0,-?
-?,-?
x1
-?,-?
-?,0
-?,2
1,1
1,1
1,1
- ?,1
-?,0
-?,2
-x1
-?,0
-?,2
1,1
1,1
1,1
-?,-?
-?,2
- ?,1
-?,0
x2
-?,-?
- ?,1
-?,2
-?,0
-?,2
-?,0
1,1
1,1
1,1
-x2
-?,-?
-?,2
-?,0
1,1
1,1
1,1
- ?,1
-?,2
-?,0
(x1 or -x2)
-?,-?
-?,-?
0,-?
-?,-?
-?,-?
- ?,1
2,-?
2,-?
0,-?
(-x1 or x2)
-?,-?
-?,-?
2,-?
-?,-?
-?,-?
- ?,1
2,-?
0,-?
0,-?
default
1,-?
1,-?
0,0
1,-?
1,-?
1,-?
1,-?
1,-?
1,-?
2

• Proof sketch
• Playing opposite literals (with any probability)
  is unstable because the other player will
  prefer to play a variable move . For example, if
  the row player plays x1 then the column player
  will play x2 which would prompt the first player
  to change his/her move to default. (Note as soon
  as one player plays default, the other players
  best response is to also play default, leading to
  the Nash equilibrium at (default, default).)
• If you play literals (with probabilities), you
  should make sure that
• for any clause, the probability of the literal
  being in that clause is high enough, and
• for any variable, the probability that the
  literal corresponds to that variable is high
  enough
• (otherwise the other player will play this
  clause/variable and hurt you)
• A player wont play any literal with a
  probability greater than 1/n because the other
  player will prefer to play a variable strategy
  which has a large negative payoff for the first
  player. For example if the row player plays x1
  with a probability of 2/3 and x2 with a
  probability of 1/3 then the column player would
  be motivated to play x2 which is a big hit to the
  row players payoff.
• So equilibria where both randomize over literals
  can only occur when both randomize over same SAT
  solution
• A player would not play a variable strategy
  because the other player would play the default
  strategy which results in a negative payoff for
  the first player . The first player would then
  decide to play the default move with a zero
  payoff for both players.
• Similarly for the clause stategies.
• The default move is needed because then it is
  true that there is a satisfying assignment for
  the SAT formula iff there is more than one mixed
  Nash equilibrium in the game. To see why, note
  that if there is no satisfying assignment, then
  any mixed strategy that one of the players uses
  would violate one of the clauses and would
  motivate the other player to play a clause move
  which leads to the default Nash the default Nash
  is the only Nash and starting from any state, a
  sequence of best replies will lead us there. If
  there is a satisfying assignment, then not only
  will (default,default) be a Nash, but the mixed
  strategy profile where each player puts
  probability 1/n on each literal of the satisfying
  assignment will also be a Nash.

3
Complexity of mixed-strategy Nash equilibria with
certain properties

• This reduction implies that there is an
  equilibrium where players get expected utility 1
  each iff the SAT formula is satisfiable
• Any reasonable objective would prefer such
  equilibria to 0-payoff equilibrium
• Corollary. Deciding whether a good equilibrium
  exists is NP-hard
• 1. equilibrium with high social welfare
• 2. Pareto-optimal equilibrium
• 3. equilibrium with high utility for a given
  player i
• 4. equilibrium with high minimal utility
• Also NP-hard (from the same reduction)
• 5. Does more than one equilibrium exists?
• 6. Is a given strategy ever played in any
  equilibrium?
• 7. Is there an equilibrium where a given strategy
  is never played?
• (5) weaker versions of (4), (6), (7) were known
  Gilboa, Zemel GEB-89
• All these hold even for symmetric, 2-player games

4
Counting the number of mixed-strategy Nash
equilibria

• Why count equilibria?
• If we cannot even count the equilibria, there is
  little hope of getting a good overview of the
  overall strategic structure of the game
• Unfortunately, our reduction implies
• Corollary. Counting Nash equilibria is P-hard!
• Proof. SAT is P-hard, and the number of
  equilibria is 1 SAT
• Corollary. Counting connected sets of equilibria
  is just as hard
• Proof. In our game, each equilibrium is alone in
  its connected set
• These results hold even for symmetric, 2-player
  games