Equilibrium Concepts in Two Player Games

1
Equilibrium Concepts in Two Player Games

• Kevin Byrnes
• Department of Applied Mathematics Statistics

2
Nash Equilibria

• A set of strategies (x,y) in a two player game
  is a Nash equilibrium point if
• f(x,y) f(x,y) for all x in S1
• g(x,y)g(x,y) for all y in S2

3
Nash Equilibria

• More generally, a set of strategies (x1,,xn)
  in an n player game is a Nash equilibrium point
  if
• f1(x1,,xn)f1(x,...,xn) for all x in S1
• fn(x1,,xn)fn(x1,,x) for all x in Sn

4
Nash Equilibria

• The key properties of a Nash equilibrium are that
  it is enforceable and guaranteed to exist under
  certain reasonable assumptions.

5
Nash Equilibria

• Enforceability comes from the fact that every
  player is unilaterally maximizing his payoff, ie
  no player can single-handedly deviate to do
  better.
• To see why it must exist, consider the following
  sketch of a proof

6
Nash Equilibria

• Lemma 1 If f1,,fn are concave, then the best
  response sets are convex
• Lemma 2 If f1,,fn are also continuous and the
  strategy spaces are compact, then the best
  response sets form an upper hemi continuous
  correspondence

7
Nash Equilibria

• Existence Theorem (Nash) Assuming that the fi
  are continuous and concave, and that the strategy
  spaces Si are compact, then a Nash equilibrium
  exists.
• Proof By Lemmas 1 and 2 the BRi(.) are upper
  hemi continuous correspondences on compact sets.
  By Kakutanis Fixed Point Thm. we then know that
  we must have a fixed point.

8
Nash Equilibria

• Now let us consider a special type of game,
  namely a bimatrix game (ie both players have a
  finite number of pure strategies and payoff
  matrices)
• Let A be player 1s payoff matrix, so
  Aijf(si,sj), where si is a pure strategy in S1,
  and sj is a pure strategy in S2.
• Let B be player 2s payoff matrix, so Bijg(si,sj)

9
Nash Equilibria

• Now let xi denote the probability with which
  player 1 plays si, and let yj denote the
  probability with which player 2 plays sj. Then a
  Nash equilibrium is a pair of strategies (x,y)
  such that x satisfies (i) and y satisfies (ii),
  where we define (i) and (ii) as

10
Nash Equilibria

• Maxx xTAy
• Subject to x1xm1
• x0
• Maxy xTBy
• Subject to y1yn1
• y0

11
Nash Equilibria

• By our existence proof, we know that such
  equilibria exist, the question is, how can we
  find them?

12
Nash Equilibria

• By our existence proof, we know that such
  equilibria exist, the question is, how can we
  find them?
• It turns out that Vorobev, Kuhn, Lemke, and
  Howson (inter alia) have proposed algorithms for
  finding Nash equilibrium points for the special
  case of bimatrix games. Computing such
  equilibria may be expensive, however. Thus we
  shall now focus on a key geometric result of
  Mangasarian that tells us which equilibrium
  points we really need to generate.

13
The Geometry of Equilibria

• For a bimatrix game, finding an equilibrium point
  is equivalent to simultaneously solving problems
  (i) and (ii). But each of these is just an LP.

14
The Geometry of Equilibria

• For a bimatrix game, finding an equilibrium point
  is equivalent to simultaneously solving problems
  (i) and (ii). But each of these is just an LP.
• Now recall that to solve a single LP, we just
  need to look at the extreme points of the
  feasible region, could we be so lucky here?

15
The Geometry of Equilibria

• Before proceeding, it is useful to note that if a
  pure strategy Nash equilibrium exists in a
  bimatrix game, it may be found in a
  straightforward fashion.
• Consider the following game

16
The Geometry of Equilibria

• A yellow box indicates the row players best
  response to a given column strategy. A red box
  indicates the column players best response to a
  given row strategy.

17
The Geometry of Equilibria

• A yellow box indicates the row players best
  response to a given column strategy. A red box
  indicates the column players best response to a
  given row strategy.

18
The Geometry of Equilibria

• A Nash equilibrium exists at the intersection of
  any of these two best responses.

19
The Geometry of Equilibria

• First recall problems (i) and (ii)
• (i) Maxx xTAy
• Subject to eTx1
• x0
• (ii) Maxy xTBy
• Subject to dTy1
• y0
• Where e and d are the appropriate vectors of all
  1s

20
The Geometry of Equilibria

• Now let w equal xTAy, and let z equal xTBy
  for a specific (x,y) solution of (i) and (ii).
  Then we have the following
• Equivalence Theorem A necessary and sufficient
  condition that (x,y,w,z) be a solution of (i)
  and (ii) is that it is a solution of the
  programming problem
• (iii) Maxx,y,w,z xT(AB)y-w-z(x,z) is in S,
  and (y,w) is in T

21
The Geometry of Equilibria

• Where
• S(x,z)BTx-zd0
• T(y,w)Ay-we0
• Note that S and T are both convex polyhedral
  sets.

22
The Geometry of Equilibria

• Now observe that
• (iv) xT(AB)y-w-z0
• In fact, by the Equivalence Theorem, any set of
  (x,y,w,z) that satisfy (iv) with (x,z) in S
  and (y,w) in T satisfy (iii).

23
The Geometry of Equilibria

• Now we shall define an extreme equilibrium point
  (x,y,w,z) as a point that satisfies (iv), and
  for which (x,z) is a vertex of S, and (y,w)
  is a vertex of T.
• Observe that by definition, there exist only a
  finite number of extreme equilibrium points, as S
  and T only have a finite number of extreme points.

24
The Geometry of Equilibria

• Lemma (Mangasarian) All equilibrium points of a
  bimatrix game may be expressed as convex
  combinations of some extreme equilibrium points.
• Proof Let (x,y,w,z) be a solution of (iii).
  Now if we set yy, and ww, then (iii) reduces
  to a linear programming problem in x and z.
  This implies, by the extreme point
  characterization of all solutions of an LP, that
  all solutions (x,y,w,z) must be convex
  combinations of some subset U of S.

25
The Geometry of Equilibria

• Thus each vertex (x,z) of U is a solution of our
  modified (iii), and so satisfies (iv)
• (v) xT(AB)y-w-z0
• In a similar fashion, we see that (y,w) must
  have been a convex combination of vertices in V,
  a subset of T. So (v) is equal to
• (vi) xT(Ay-we)yT(BTx-zd)0, for (x,z) in U

26
The Geometry of Equilibria

• Now note that x0, y0, and Ay-weBTx-zdU. So (vi) implies
• yT(BT-zd)0 for (x,z) in U

27
The Geometry of Equilibria

• Now note that x0, y0, and Ay-weBTx-zdU. So (vi) implies
• yT(BT-zd)0 for (x,z) in U
• Since (y,w) is a convex combination of points
  in V, (vii) implies that
• yT(BTx-zd)0 for (x,z) in U and some (y,w) in V

28
The Geometry of Equilibria

• Similarly, we have (can show) that
• xT(Ay-we)0 for (y,w) in V and some (x,z) in U
• This gives us that
• xT(Ay-we)yT(BTx-zd)0
• ie xT(AB)y-w-z0
• for some (y,w) in V and some (x,z) in U
• Which proves the claim.

29
NonInferior Equilibria

• A set of strategies (x,y) in a two player game
  is a noninferior equilibrium point if
• There does not exist x,y in S1XS2such that
• f(x,y)f(x,y) and g(x,y) g(x,y)
• or f(x,y) f(x,y) and g(x,y)g(x,y)

30
NonInferior Equilibria

• An example of noninferior equilibria
• Suppose that we are given the following (x,y)
  pairs and their associated payoffs

31
NonInferior Equilibria

• The noninferior points are (x1,y1) and (x2,y2)

32
NonInferior Equilibria

• The noninferior points are (x1,y1) and (x2,y2)

33
NonInferior Equilibria

• For a bimatrix game, a noninferior set of
  strategies (x,y) is a pair that satisfies the
  multiobjective problem
• (x) Maxx xTAy, Maxy xTBy
• Subject to x1xm1
• y1yn1
• x0
• y0

34
NonInferior Equilibria

• Not every Nash equilibrium is noninferior
  however, for example, the Prisoners Dilemma

35
NonInferior Equilibria

• Here the unique NE is (Confess, Confess), but its
  payoff (-2,-2) is strictly inferior to (-.5,-.5)
  that of (Dont Confess, Dont Confess)

36
NonInferior Equilibria

• Here the unique NE is (Confess, Confess), but its
  payoff (-2,-2) is strictly inferior to (-.5,-.5)
  that of (Dont Confess, Dont Confess)

37
NonInferior Equilibria

• The previous example demonstrates that
  noninferiority may be a better solution concept
  than Nash equilibria, as it is a true maximizer.
  
• The downside is that a noninferior equilibrium
  may note be enforceable. In our previous
  example, both players could benefit by
  unilaterally deviating from the noninferior
  equilibrium of (Dont Confess, Dont Confess)
• This begs the question, can we perturb the
  payoffs in a nice manner to impose
  enforcability?

38
Future Directions

• 1) Can we find an appropriate penalty to
  transform the noninferior equilibria of some or
  all games into enforceable ones?
• 2) Suppose that a given player knows only one of
  A or B, is there an evolutionary strategy that
  maximizes his expected payoff if the game is
  repeated infinitely often? What about finitely
  often?