1. Enumerating All Nash Equilibria for Two-person Extensive Games
Huang Wan
London School of Economics
2019/2/24

2. Games in Extensive Form
Games in extensive form are represented by trees, with players moves corresponding to tree edges and information sets corresponding to tree nodes. A player I has perfect recall iff all nodes in the same information set of player I have the same own former paths. There are two strategies L and R for player 2 and four strategies ac,ad,bc,bd for player 1.
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3. Sequence Form Hi - set of information sets of player i
h - information set
Ch - set of moves at h
Si - set of sequences of player i
Any sequence is either the empty sequence or uniquely given by its last move at the information set of that player.
Bernhard introduced the sequence form for a game tree. The sequence form is analogous to an strategic form. The counterpart of a strategy of an strategic form is a sequence in the sequence form. The good thing for the sequence form is that the number of sequences of a player is polynomial in the size of the game tree, while the number of strategies in the strategic form is exponential in the size of the game tree.
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4. Example: sequence form
The first decision point is a square (rectangle) in the game tree, which means the first move is made by the nature. There are 2 information sets for player 1. Sequences a and b are uniquely determined by the last moves at the left information set; c and d are determined by the right one. There is only one information set for the second player. Sequences L and R are given by the moves L and R from that information set.
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5. Property of realization plans ( von Stengel, 1996)
Suppose x and y are realization plans for player 1 and player 2, respectively. Then
A realization plan sets a probability to each sequence.
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6. Example: sequence form
Here x and y are the realization plans for player 1 and 2, respectively. According to the property, we have Ex=e and Fy=f, where E and F are the constraint matrices. Each column of E and F corresponds to a sequence and each column corresponds to a information set.
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7. Linear Programs
By the definition of best response, (x, y) is an equilibrium iff it is in the solution of the pair of linear programs (LPs):
The objective function of the left LP means that x should be a best response for the given y; and the right one means y should be a best response for the given x. The subjective equations and inequalities ensure x and y are realization plans for the players.
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8. Linear Programming Duality
Primal Dual
P 1
P 2
On the right column of this form is the dual of the pair of the LP. The first row for player 1 and second for player 2.
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9. Complementary Slackness
Theorem (von Stengel, 1996) The pair (x, y) of realization plans defines an equilibrium iff
there are vectors u, v, such that
In the last line there are the complementarity conditions.
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10. Removing Redundancy
Terminal sequence - A sequence is terminal iff there is no move extending to a longer sequence.
Independent sequence - A sequence of player 1 is called independent (in a solution of Ex=e) iff its realization probability is a free variable in that solution. Similarly for sequences of player 2.
-- The independence of a sequence relates to which free variables are chosen for the solution of Ex=e.
The next step is supposed to be, drawing two polyhedra, each for one player, according to the equations and inequalities in the theorem. And find out all the points in the polyhedra that satisfies the complementarity conditions. These points represents all the equilibria of the game. But before doing that, we’d like to eliminate some of the redundant variables as well as some redundant constraints. In order to do that, we observe the sequence of the game and define the terminal sequence and the independent sequent.
A terminal sequence lead to a leaf. But not all sequences that lead to a leaf are terminal. Some non-terminal sequences can still lead to leaves.
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11. Removing Redundancy
In our algorithm we only choose the realization plans of terminal sequences as free variables (thus only terminal sequences can be independent). We can do this because:
Lemma 1 For every non-terminal sequence , there are some terminal sequences , s.t.
Proof. (by contradiction)
Therefore all the non-negativity constraints of the non-terminal sequences are redundant if all the non-negativity constraints of the terminal sequences are satisfied.
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12. Removing Redundancy Three kinds of sequences:
Terminal sequences whose realization probabilities xI are independent
Terminal sequences whose realization probabilities xD are dependent, which can be expressed as p2+P2xI
Non-terminal sequences whose realization probabilities xN are dependent, which can be expressed as p1+P1xI, and for which the non-negativity constraints are redundant
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13. Removing Redundancy Realization plans for both players: Player 1
where
p_1, p_2, q_1, q_2 are certain vectors and P_1,P_2, Q_1, Q_2 are certain matrices. I_m-k and I_n-l are the identity matrices.
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14. Removing Redundancy Primal and Dual after removing redundancy :
Primal Dual
P 1
P 2
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15. Removing Redundancy
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16. Removing Redundancy LCP after removing redundancy:
Theorem 1 The pair (x,y) of realization plans defines an equilibrium iff there are vectors u,v, such that
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17. Removing Redundancy 2019/2/24
This theorem is analogues to the last one by bernhard (von Stengel 1996)
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18. Finding All Equilibria
Consider the polyhedra
Finding all equilibria of the game can
be converted to an inspection of all
vertices of and .
Now we can move to the step drawing the polyhedra. The polyhedra, each for one player, are defined by the constrains after the redundancy removal.
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19. Finding All Equilibria
Label of points
Since redundant constraints will not be considered, we can re-index only the terminal sequences. Suppose there are s terminal sequences for player 1 and t for player 2. Let
Labels are used in Lemke-Howson algorithm. We use the labels here to find out the points that satisfy all the complementarity conditions.
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20. Finding All Equilibria
The labels of (xI,v) in D1 and (yI, u) in D2 are given by:
Each label corresponds to a binding constraint.
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21. Finding All Equilibria
Lemma 2 For a two-person extensive form game (A,B), a realization plan (x,y) is an equilibrium iff (x,y) is a pair of points in D1 and D2 that has full labels 1,…,s+t.
This lemma is equivalent to Theorem 1.
The pair of points that has full labels satisfies the complementarity conditions.
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22. Finding All Equilibria
Lemma 3 Suppose y* is a point in D2 which has l* labels. Then (x,y*) is an equilibrium iff x is in the set of convex combination of vertices or points in the extreme rays which have all the rest labels in D1. Similarly for points x* in D1.
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24. Finding All Equilibria
Lemma 4 Suppose (x*,y) represents an equilibrium where y is a point in D2 and x* is in the convex combination of vertices and points in extreme rays or in the convex combination of points in extreme rays in D1, then there is a point x’, which is in the convex combination of certain vertices, such that (x’,y) represents the same equilibrium as (x*,y).
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26. There are the other two cases when x
There are the other two cases when x* is in the convex combination of some vertices and some points in the extreme rays, and the case when x* is in the convex combination of some points in the extreme rays.
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27. The Algorithm
Enumerating the vertices of a polyhedron by linear inequalities: one recent method is lrs (Avis and Fukuda (1992) ).
Convex Combination of full label pairs of vertices.
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28. References.
D.Avis (2002), lrs: A revised implementation of the reverse search vertex enumeration algorithm.
V.Chvatal (1983), Linear Programming. Freeman, New York.
H.W.Kuhn (1961), An algorithm for equilibrium points in bimatrix games. Proc. National Academy of Sciences of the U.S.A. 47,
C.E.Lemk and J.T.Howson, Jr. (1964), Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12,
B.von Stengel (1996), Efficient computation of behaviour strategies. Games and Economic Behaviour 14,
B.von Stengel (2002), Computing equilibria for two-person games. Handbook of Game Theory, Volume 3,
G.M.Ziegler (1995), Lectures on Polytopes. Graduate Texts in Mathematics, Vol. 152, Springer, New York.
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