Game Theory

1
Game Theory

• Used to analyze situations when rational action
  depends on other agents actions.
• Equilibrium is an important game theory concept
• Useful definition depends on the model.
• Nash Equilibrium important and well known.
• At equilibrium, no one can gain by unilaterally
  changing strategy.

Prisoners Dilemma
Nash equilibrium
2
Motivation for Graphical Games

• Tabular representation too large.
• n players, m actions n x mn entries
• Probability distributions also have huge tabular
  representations.
• Belief Networks attempt to capture locality of
  interaction.
• Instance of locality in games

• Payoff depends on at most 3 neighbors
• Tabular 10.310
• Locality 6.344.33

or
or
3
Graphical Games
Graphical model for the factory, shopping mall,
residential complex game

• If k-1 is an upper bound on neighborhood size
  then mk entries.
• Undirected graph corresponds to an assumption
  about symmetry of interaction.
• Can consider directed models too.
• Get exponential savings if k n.

4
Inference vs Equilibriums

• Exact P-complete
• Poly time exact algo for trees
• Basic message passing algo generalized in 2 ways
• Junction tree
• Loopy belief propagation

• Exact Open question (like factoring)
• Poly time exact algo for trees
• Similar generalizations attempted
• No real counterpart of junction tree
• NashProp

5
TreeNash algorithm

• Abstract algorithm Kearns et. al. two
  realizations.

u1
u2
Upstream Pass
Downstream Pass
v
T(w,v)
w

• Message T(w,v) is a table with continuous
  indices and is 1 iff given Ww, there is an
  upstream Nash at V in which Vv.
• Not clear how to implement.
• Two solutions
• discretize and obtain poly time algorithm.
• work with clever (but exponential)
  representation of sets where T(w,v) is 1.

6
Two realizations

• Approximate
• Players play strategies that are multiples of t.
• T(w,v) is just a 1/t by 1/t matrix.
• To compute e-Nash, need t of the order
  poly(e,2-k) running time poly(n,1/e,2k).
• Exact
• T sets are always disjoint unions of axis
  parallel rectangles.
• of rectangles 2a3b (ainternal nodes,
  bleaves)
• Allows computation of every Nash equilibrium.
• Also have a polynomial time algo Littman at.
  al. to compute exact equilibrium in trees but it
  throws out potential equilibriums.

7
Function minimization and CSP

• Can handle general graphs.
• Define Regreti max amount which Pi can gain by
  unilaterally changing strategy.
• Computing Nash equilibrium is same as minimizing
  sum of regrets.
• Greedy hill climbing with random restarts
  proposed in Vickrey Koller
• e-Nash can be cast as a CSP.
• Constraint for player V says that projection of
  global strategy to neighbors of V is restricted
  to the set where Regreti e.
• Discretize strategy space and use discrete CSP
  techniques.

8
NashProp algorithm

• Two stage algorithm for general graphs Ortiz
  Kearns
• table passing stage
• assignment passing stage
• Table passing is in rounds tables provably
  converge as we complete more rounds.
• Search space for Nash equilibriums gets
  (potentially) reduced.
• Assignment passing phase is a backtracking search
  through reduced search space.
• No guarantees for the running time.

9
GRAPHICAL GAMES A SURVEY
Ambuj Tewari (ambuj_at_cs)CS281A Poster Session

• References
• Kearns et. al. Graphical models for game
  theory, UAI 2001.
• Littman et. al. An efficient exact algorithm
  for solving tree-structured graphical games, NIPS
  2002
• Ortiz Kearns Nash propagation for loopy
  graphical games, to appear
• Vickrey Koller Multi-agent algortihms for
  solving graphical games, AAAI 2002