4.3 Examples
4.3.4 Young’s Example
Consider the 3×3 game given in Figure 2(a), taken from Young (1993). Oyama (2002) shows by direct computation that (2,2) is linearly absorb- ing and globally accessible for a small degree of friction. In fact, (2,2) is a strict MP-maximizer with a strict monotone potential function that is supermodular (Figure 2(b)), while the original game is not supermodular (for any ordering of actions). Therefore, our results, Theorems 4.1 and 4.2, also apply to this game.
Note that (1,1) is stochastically stable (Young (1993)), while it is neither absorbing nor globally accessible when the friction is small.
0 1 2 0 1 2 6, 6 0, 5 0, 0 5, 0 7, 7 5, 5 0, 0 5, 5 8, 8
(a) Original game
0 1 2 0 1 2 6 5 0 5 7 5 0 5 8
(b) Monotone potential function
Figure 2: Young’s example
5
Binary Supermodular Games
In this section, we restrict our attention to supermodular games with two actions for each player, where Ai = {0,1} for all i ∈ I. Note that in this
case, the stochastic dominance order reduces to the following simple form:
φ - ψ if and only if φi1(t) ≤ ψi1(t) for all i ∈ I and all t ≥ 0. Denoting pj =xj1, we define theincentive function di: [0,1]N →Rfor player iby
di(p1, . . . , pN) =ui(1, x−i)−ui(0, x−i).
In the following, we identifya= (ai)i∈I ∈A withp= (p1, . . . , pN)∈[0,1]N
such that pi = 0 if ai = 0 and pi = 1 if ai = 1. We assume that action
profiles0, where all players play 0, and1, where all players play 1, are strict Nash equilibria, i.e.,
di(0)<0< di(1)
for all i. We further assume that di is nondecreasing in each pj (j 6=i) so
characterizations for the strict Nash equilibrium 1 to be globally accessible and to be absorbing (or, equivalently, linearly absorbing), respectively. By reversing the orders of actions, the results can be applied to the other Nash equilibrium0. The subsequent subsections then consider three subclasses of binary supermodular games.
For a feasible pathφ, denote
∆Viθ(φ)(t) =Vi1(φ)(t)−Vi0(φ)(t)
= (1 +θ)
∫ ∞
t
e−(1+θ)(s−t)di(φ(s))ds.
We write the superscript θ of ∆V to specify the discount rate. Note that
∆Vθ is well-defined also for θ = 0. Recall from Lemma 3.1 that if φ -
ψ, then ∆Viθ(φ)(t) ≤ ∆Viθ(ψ)(t) for all i ∈ I and all t ≥ 0 due to the supermodularity.
5.1 General Results
ForT= (Ti)i∈I ∈RN+, let φuT be the feasible path given by
(φuT)i1(t) = { 0 ift < Ti 1−e−(t−Ti) ift≥T i, (5.1) which starts at0 and converges to1. AlongφuT, agents in population i∈I
start choosing action 1 at timeTi.
Denote ¯R+ =R+∪ {∞}. For T= (Ti)i∈I ∈R¯N+, let ψTd be the feasible path given by (ψTd)i1(t) = { 1 ift < Ti e−(t−Ti) ift≥T i fori∈S, (5.2) and (ψTd)i1(t) = 1 fori /∈S, (5.3)
whereS ={i∈I|Ti 6=∞}. Let0S be the action profile such thatichooses
0 if i∈S and 1 if i /∈S. AlongψTd, which starts at 1 and converges to 0S,
agents in populationi∈S start choosing action 0 at timeTi, while those in
populationi /∈S always play action 1.
First, we provide necessary and sufficient conditions for the state1to be globally accessible for a given degree of friction (Proposition 5.1.1) and for any small degree of friction (Proposition 5.1.2), respectively. Each condition is equivalent to the existence of a subpath of the form (5.1).
Proposition 5.1.1. Let θ > 0 be given. The strict Nash equilibrium 1 is globally accessible for θ if and only if there exists T = (Ti)i∈I ∈ RN+ such
that for all i∈I,
Proof. See Appendix.
Proposition 5.1.2. There existsθ >¯ 0such that the strict Nash equilibrium
1 is globally accessible for all θ ∈ (0,θ¯) if and only if there exists T = (Ti)i∈I ∈RN+ such that for all i∈I,
∆Vi0(φuT)(Ti)>0.
Proof. See Appendix.
Next, we provide necessary and sufficient conditions for the state1to be absorbing for a given degree of friction (Proposition 5.1.3) and for any degree of friction (Proposition 5.1.4), respectively. Each condition is equivalent to the nonexistence of a superpath of the form (5.2)–(5.3) with0S being a Nash
equilibrium of the stage game.
Proposition 5.1.3. Let θ > 0 be given. The strict Nash equilibrium 1 is absorbing for θ if and only if for any T= (Ti)i∈I ∈R¯N+ such that S ={i∈ I|Ti 6= ∞} is nonempty and 0S is a Nash equilibrium, there exists i ∈ S
such that
∆Viθ(ψdT)(Ti)>0.
Proof. See Appendix.
Proposition 5.1.4. The strict Nash equilibrium1is absorbing for all θ >0
if and only if for any T= (Ti)i∈I ∈ R¯N+ such that S ={i∈I|Ti 6=∞} is
nonempty and 0S is a Nash equilibrium, there exists i∈S such that
∆Vi0(ψdT)(Ti)≥0.
Proof. See Appendix.
5.2 Unanimity Games
This subsection considers N-player unanimity games. The stage game is given by ui(a) = yi ifa=0 zi ifa=1 0 otherwise, (5.4)
whereyi, zi >0. The incentive function for playeriis then given by
di(p1,· · · , pN) =zi ∏ j6=i pj−yi ∏ j6=i (1−pj).
For T= (Ti)i∈I∈RN+, let πi(T) = ∫ ∞ Ti e−(t−Ti)∏ j6=i [ 0∨{ 1−e−(t−Tj)}] dt = ∫ ∞ maxjTj e−(t−Ti)∏ j6=i { 1−e−(t−Tj)} dt, (5.5) and ρi(T) = ∫ ∞ Ti e−(t−Ti)∏ j6=i { 1∧e−(t−Tj)} dt. (5.6) 5.2.1 Global Accessibility
For a feasible pathφuT defined by (5.1) with a given T= (Ti)i∈I ∈RN+, the
discounted payoff difference is given by
∆Vi0(φuT)(Ti) =ziπi(T)−yiρi(T),
so that∆Vi0(φuT)(Ti)>0 if and only ifzi/yi > ρi(T)/πi(T).
We immediately have the following from Proposition 5.1.2.
Proposition 5.2.1. Suppose that the stage game is a unanimity game given by (5.4). Then there exists θ >¯ 0 such that 1 is globally accessible for all
θ∈(0,θ¯) if and only if there exists T∈RN+ such that for all i∈I,
zi
yi
> ρi(T) πi(T)
.
Symmetrically, there existsθ >¯ 0such that 0is globally accessible for all
θ∈(0,θ¯) if and only if there exists T∈RN+ such that for all i∈I, yi zi > ρi(T) πi(T) . 5.2.2 Absorption
For a feasible pathψdT defined by (5.2) with a givenT= (Ti)i∈I∈RN+, the
discounted payoff difference is given by
∆Vi0(ψTd)(Ti) =ziρi(T)−yiπi(T),
so that∆Vi0(ψTd)(Ti)≥0 if and only ifzi/yi ≥πi(T)/ρi(T).
We have the following from Proposition 5.1.4. Observe that in this case,
Proposition 5.2.2. Suppose that the stage game is a unanimity game given by (5.4). Then 1 is absorbing for all θ > 0 if and only if for any T ∈RN+, there existsi∈I such that
zi
yi ≥
πi(T)
ρi(T)
.
Symmetrically,0is absorbing for allθ >0if and only if for anyT∈RN+,
there existsi∈I such that
yi zi ≥ πi(T) ρi(T) . 5.2.3 Two-Player Case
In the case whereN = 2, there existsT∈R2+ such that z1 y1 > ρ1(T) π1(T), z2 y2 > ρ2(T) π2(T)
if and only if z1z2 > y1y2. Therefore, by Propositions 5.2.1 and 5.2.2, 1
is absorbing and globally accessible for any small degree of friction if and only if1 has the higher Nash product over0. In the two-player case, this is equivalent to that1 risk-dominates 0.
5.2.4 Three-Player Case
WhenN ≥3, the complete characterizations given in Propositions 5.2.1 and 5.2.2 turn out to be rather complex. Here we consider three-player binary games with a symmetry between players 2 and 3. We demonstrate that even for this simple class of games, both Nash equilibria 1 and 0 may be simultaneously globally accessible states when the friction is small.
Specifically, we consider the case where
(z1/y1, z2/y2, z3/y3) = (r, s, s). (5.7) We can exploit the symmetry due to the following fact. Note here that if
Ti =Tj, then πi(T) =πj(T) and ρi(T) =ρj(T).
Lemma 5.2.3. Suppose that the stage game is given by (5.4). Then 1 is globally accessible for any small degree of friction if and only if there exists
T such that for all i∈I,
zi yi > ρi(T) πi(T) , (5.8) and zi yi ≥ zj yj ⇒ Ti≤Tj. (5.9)
Proof. It suffices to show that if there existsTthat satisfies (5.8), then there existsT′ that satisfies both (5.8) and (5.9).
Take Tthat satisfies (5.8) and define T′ by
Ti′= min
j:zj/yj≤zi/yi Tj
for each i. Note that T′
i ≤Ti for anyi.
Here we fix any i. By definition, there exists j such that Ti′ = Tj and
zj/yj ≤zi/yi. Take such aj. Note that T−j ≥T′−j and Tj =Tj′. Since T
satisfies (5.8),πj is decreasing inT−j, andρj is increasing in T−j, we have
zj yj > ρj(T) πj(T) ≥ ρj(T′) πj(T′) .
On the other hand, πi(T′) = πj(T′) and ρi(T′) = ρj(T′) since Ti′ = Tj′.
Therefore, it follows fromzj/yj ≤zi/yi that
zi yi ≥ zj yj > ρj(T ′) πj(T′) = ρi(T ′) πi(T′) ,
which completes the proof.
A direct computation utilizing Lemma 5.2.3 shows that 1 is globally accessible for a small friction if and only if there existsu≥1 such that
r < s, r > 1
3u2−3u+ 1, s >
3u2−1
3u−1 , or there existsv≥1 such that
r≥s, r >3v−2, s > 2
3v−1. The above condition is equivalent to that
r < s and r > 2
(s−1)√9s2−12s+ 12 + 3s2−5s+ 4,
or
r ≥s and r > 2 s−1.
In the game given by (5.7), 1 has the higher Nash product over 0 if
rs2 >1. A direct comparison betweenr >1/s2 and the above expressions gives the following sufficient condition in terms of Nash product.
Proposition 5.2.4. In the game given by (5.7), the Nash equilibrium with the higher Nash product is globally accessible for any small degree of friction.
Example 5.2.1. Let y1 = 6 +c > 0, y2 = y3 = 1, and z1 = z2 =z3 = 2
(see Figure 3). This game is a modified version of an example in Morris and Ui (2005, Example 1).20 If c > 0, then 0 is globally accessible for a small friction, while if c < 2√6, then 1 is globally accessible for a small friction. Therefore, if 0 < c < 2√6, the game has two globally accessible states simultaneously when the friction is small. Note that0 (1, resp.) has the higher Nash product ifc >2 (c <2, resp.).
On the other hand, one can show that if c≤0, then 1 is absorbing for any degree of friction, while ifc≥2√6, then 0 is absorbing for any degree of friction. 0 1 0 1 0 6 +c, 1, 1 0, 0, 0 0, 0, 0 0, 0, 0 0 1 0 1 1 0, 0, 0 0, 0, 0 0, 0, 0 2, 2, 2
Figure 3: Multiple globally accessible states