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Stat 155 Game theory, Yuval Peres Fall 2004
Lectures 29,30,
Proof of Brouwer’s fixed point theorem (from the no retraction theorem): Recall that we are given a continuous map T : K → K, with K a closed, bounded and convex set. Suppose that T has no fixed point. Define F : K → ∂K as follows. For each x ∈ K, we draw a line segment from T (x) through x until it meets ∂K. We set F (x) equal to this point of intersection. (note that, in the case that T (x) ∈ ∂K, we set F (x) equal that point of intersection of the line segment with ∂K which is not equal to T (x)). In the case of the domain K = {(x, y) ∈ R^2 : x^2 +y^2 ≤ 1 }, the map F may be written explicitly in terms of T. With some checking, it follows that F : k → ∂K is continuous. Thus, F is a retraction of K - but none exists, by the no retraction theorem. This contradiction establishes the theorem.
Potential games: We now discuss a collection of games called potential games. These are k-player general sum games that have a special feature: let Fi(s 1 , s 2 ,... , sk) denotes the payoff to player i if the players respectively adopt the pure strategies s 1 , s 2 ,... , sk. In a potential game, we suppose that there is a function Ψ : S 1 ×... Sk → R, defined on the product of the players’ strategy spaces, and such that
Fi(s 1 ,... , si− 1 , s˜i, si+1,... , sk) − Fi(s 1 ,... , sk) = ψ(s 1 ,... , si− 1 , ˜si, si+1,... , sk) − ψv(s 1 ,... , sk). (1)
We call the function ψ : S 1 ×... Sk → R the ‘potential’ function associated with the game. We have the following result:
Theorem 1 (Shmeidler-Shapley) Every potential game has a Nash equi- librium in pure strategies.
Proof: this appears after the following example.
Example of a potential game: a simultaneous congestion game: In this sort of game, the cost of using each road depends on the number of users of the road. For the road AC, it is li if there are i users, for i ∈ { 1 , 2 }, in the case of the game depicted in the figure. Note that the cost paid by a given driver depends only on the number of users, not on which user she is.
A C
B D
(r 1 , r 2 )
(l 1 , l 2 )
(m 1 , m 2 ) (n 1 , n 2 )
More generally, we may define R-valued map C on the product space of the road-index set and the set { 1 ,... , k}, so that C(j, uj ) is equal to the cost incurred by any driver using road j in the case that the total number of drivers using this road is equal to uj. Note that the vector s = (s 1 , s 2 ,... , sk) determines the usage of each road. That is, it determines ui(s) for each i ∈ { 1 ,... k}, where
ui(s) =
j ∈ { 1 ,... , k} : player j uses road i under strategy sj
for i ∈ { 1 ,... , R} (with R being the number of roads.) In the case of the game depicted in the figure, we suppose that two drivers, I and II, have to travel from A to D, or from B to C, respectively. In general, we set
ψ(s 1 ,... , sk) = −
∑^ R
r=
u ∑r (s)
l=
c(r, l).
We claim that ψ is a potential function for such a game. We show why this is so in the specific example. Suppose that driver 1, using roads 1 and 2, makes a decision to use roads 3 and 4 instead. What will be the effect on her cost? The answer is a change of ( c(3, u 3 (s) + 1) + c 3 (4, u 4 (s) + 1)
c(1, u 1 (s)) + c(2, u 2 (s))
How did the potential function change as a result of her decision? We find that, in fact,
ψ(s) − ψ(˜s) = c(3, u 3 (s) + 1) + c 3 (4, u 4 (s) + 1) − c(1, u 1 (s)) − c(2, u 2 (s))
where ˜s denotes the new joint strategy (after her decision), and s denotes the previous one. Noting that payoff is the negation of cost, we find that the change in payoff is equal to the change in the value of ψ. To show that ψ is indeed a potential function, it would be necessary to reprise this argument in the case of a general change in strategy by one of the players. Proof of Shmeidler-Shapley theorem: Choose s that maximizes ψ(s). Note that the expression in (1) is at most zero, for any i ∈ { 1 ,... , k} and any choice of ˜si. This implies that s is a Nash equilibrium.
member of S. It has characteristics function wS , given by wS (T ) = 1 if and only if S ⊆ T. In this case, the Shapley value is given by
ψi(wS ) =
|S|
if i ∈ S,
with ψi(wS ) = 0 if i 6 ∈ S. Moreover, we have that ψi(cWs) = cψi(Ws) for each c ∈ [0, ∞). Note that the glove market game has the same payoffs as w 23 + w 13 , except for the case of the set { 1 , 2 , 3 }. In fact, we have that
w 23 + w 13 = v + w 123.
In particular, we have that ψi(w 23 ) + ψi(w 12 ) = ψi(v) + ψi(w 123 ). If i = 1, then 0 + 1/2 = ψ 1 (v) + 1/ 3 ,
while, if i = 3, then 1 /2 + 1/2 + ψ 3 (v) + 1/ 3.
Hence, ψ 3 (v) = 2/3, and ψ 1 (r) = ψ 2 (r) = 1/6. This means that player I has two-thirds of the arbitration value, while player II and III have one-third between them. Example: the four stockholders. Four people own stock in ACME. Player i holds i units of stock, for each i ∈ { 1 , 2 , 3 , 4 }. Six shares are needed to pass a resolution at the board meeting. How much is the position of each player worth in the sense of Shapley value? Note that
1 = v 1234 = v 24 = v 34 ,
with v = 1 on any 3-tuple, and v = 0 in each other case. We will assume that the value v may be written in the form
v =
∅6=S
cS wS.
Later, we will see that there always exists such a way of writing v. For now, however, we assume this, and compute the coefficients cS. To do so, note that 0 = v 1 = c 1
(we write c 1 for c{ 1 }), and so on. Similarly,
0 = c 2 = c 3 = c 4.
Also, 0 = v 12 = c 1 + c 2 + c 12 ,
implying that c 12 = 0. Similarly,
c 13 = c 14 = 0.
Next, 1 = v 24 = c 2 + c 4 + c 24 ,
implying that c 24 = 0. Similarly, c 34 = 1. We have that
1 = v 123 = c 123 ,
while 1 = v 124 = c 24 + c 124 ,
implying that c 124 = 0. Similarly, c 134 = 0, and
1 = v 234 = c 24 +c 34 +c 123 +c 124 +c 134 +c 234 +c 1234 = 1+1+1+0+0−1+c 1234 ,
implying that c 1234 = −1. Thus,
v = w 24 + w 34 + w 123 − w 234 − w 1234 ,
whence ψ 1 (v) = 1/ 3 − 1 /4 = 1/ 12 ,
and ψ 2 (v) = 1/2 + 1/ 3 − 1 / 3 − 1 /4 = 1/ 4 ,
while ψ 3 (v) = 1/4, by symmetry with player 2. Finally, ψ 4 (v) = 5/12. Probabilistic interpretation of Shapley value: Suppose that the play- ers arrive at the board meeting in a uniform random order. With the arrival of precisely one stockholder, the coalition present in the board-room be- comes effective. Then the Shapley value of a given player is the probability that that player is the one to make the existing coalition effective, where the players enter the board-room in this particular random fashion. We will write this statement in symbols and prove it later.
