A New Interpretation of the Nash Bargaining Solution: Fictitious Play

A new interpretation of the Nash bargaining solution: ﬁctitious play∗

Younghwan In†

This version: January 2015

Abstract

We provide a new interpretation of the Nash bargaining solution, using fictitious play. Based on the finding that the Nash demand game has the fictitious play property and that almost every fictitious play process and its associated belief path converge to a pure-strategy Nash equilibrium in the Nash demand game (In, 2014), we present two initial demand games which exactly and approximately implement the Nash bargaining solution.

JEL Classiﬁcation: C71, C72, C78.

Keywords: Nash bargaining solution, Nash demand game, ﬁctitious play.

∗ I thank Karl Schlag and Roberto Serrano for their valuable comments. † KAIST College of Business, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea. Tel: +82-2-958-3314. fax: +82-2-958-3604. e-mail: [email protected] 1 Introduction

We provide a new interpretation of the Nash (1950) bargaining solution, using fictitious play (Brown, 1951). Nash (1953) stated that the non-cooperative and axiomatic approaches to bargaining problems are complementary, and presented the perturbed Nash demand game as a non-cooperative game to yield as its equilibrium the Nash bargaining solution (or the Nash solution, in short). This idea of relating axiomatic solutions to equilibria of non-cooperative games is now known as the Nash program. An important work in this was done by Binmore et al. (1986). They showed that the subgame-perfect equilibrium of the Rubinstein’s (1982) alternating-offers game approaches the Nash solution as the friction vanishes. See also Carlsson (1991) and Güth et al. (2004). Serrano (2005) provided a recent survey on the Nash program. There have been attempts to provide evolutionary or learning founda- tions of the Nash solution too. Several authors have shown how a particular equilibrium is selected among the many Nash equilibria of the Nash demand game in evolutionary or learning models. Young (1993, 1998) presented a model where players make their demands by choosing best replies based on an adaptive play process with incomplete sampling and showed that the stochastically stable divisions converge to the Nash division. Santamaria- Garcia (2009) showed similar convergence results using a matching frame- work similar to Kandori et al.’s (1993). Binmore et al. (2003) showed that the Nash division is the unique stochastically stable equilibrium under best- response dynamics. See also Skyrms (1994), Ellingsen (1997), Oechssler and Riedel (2001), and Agastya (2004). Little attention has been paid to the fictitious-play dynamics of the Nash demand game. When each player chooses its demand according to the fictitious play (Brown, 1951) in a repeated Nash (1953) demand game, in every period following the initial period each player chooses its best response to the empirical distribution of its opponent’s past choices. In a companion paper (In, 2014), we showed that the Nash demand game has the fictitious play property and that almost every fictitious play process and its associated

1 belief path converge to a pure-strategy Nash equilibrium in the Nash demand game. The limit to which the fictitious play process converges depends on the initial demands and therefore the fictitious play process alone cannot select a particular equilibrium. We consider two one-shot games in each of which the players choose their initial demands, which we call initial demand games. The first initial demand game, where we use the limit of average payoffs to rank the infinite payoff sequences, exactly implements the Nash solution. Thanks to this result, we can state that the “Nash” solution is the “Nash” equilibrium of the “Nash” demand game with fictitious play. The second initial demand game, where we use the discounted average of the infinite payoff sequences to rank them, approximately (àla Radner’s (1980) -equilibrium) implements the Nash solution. As the players make a more accurate comparison of payoffs and become more patient accordingly, the set of -equilibria in the neighborhood of the Nash division shrinks and the only -equilibrium remaining is the Nash division.

2 Preliminaries

Two players, 1 and 2 play the Nash demand game in each period t ∈ {0, 1, 2,...}. The size of the surplus to be divided is normalized to be 1. In each period players 1 and 2 simultaneously announce their demands x and y respectively, where x, y ∈ [0, 1]. If x + y ≤ 1, they receive u(x) and v(y) respectively in that period. Otherwise, they receive u(0) and v(0) respectively. The utility functions u and v are continuous, strictly increasing, concave, and normalized such that u(0) = v(0) = 0 and u(1) = v(1) = 1. As is well known, there are a multitude of Nash equilibria in the one-shot Nash demand game: (1, 1) and all (x, y) such that x + y = 1 are the pure-strategy Nash equilibria. From period t = 1, each player believes that its opponent chooses its mixed strategy corresponding to the empirical distribution of its past choices and chooses its own best response to this belief, myopically in each period.

We deﬁne relative frequencies ft(x) and gt(y) recursively. For t ∈ {0, 1, 2,...}

2 and x, y ∈ [0, 1],

 tf (x)+1  t if player 1 chooses x in period t f (x) ≡ t+1 t+1 tft(x)  t+1 otherwise, and

 tg (y)+1  t if player 2 chooses y in period t g (y) ≡ t+1 t+1 tgt(y)  t+1 otherwise, where f0(x) ≡ g0(y) ≡ 0 for any x and y. For t ∈ {1, 2, 3,...}, we call ft : [0, 1] → [0, 1] and gt : [0, 1] → [0, 1] empirical distributions in period t. We call the following choice rule the ﬁctitious play: for any t ∈ {1, 2, 3,...}, players 1 and 2 choose xt and yt respectively such that

X xt = max ( arg max gt(y)u(x)I[x+y≤1](x, y) ) and (1) x y:gt(y)>0 X yt = max ( arg max ft(x)v(y)I[x+y≤1](x, y)), (2) y x:ft(x)>0 where I is the indicator function. Note that we assume ties are broken in ∞ favor of the highest demand. We call the sequence (xt, yt)t=0 a fictitious play ∞ process if (xt, yt)t=1 are determined according to the fictitious play and the ∞ sequence (ft, gt)t=1 its associated belief path. Let (xN , yN ) be the division of the Nash (1950) solution, given u and v (we omit the dependence on u and v in the notation). That is, (xN , yN ) ≡ arg max(x,y) u(x)v(y) subject to x ≥ 0, y ≥ 0, and x + y ≤ 1. We define a function φ : [0, 1] → [0, 1] that associates each x ∈ [0, 1] with φ(x) ∈ [0, 1] as follows (see Figure 1 of In (2014)):

• φ(xN ) = yN .

• If x 6= xN , φ(x) is the solution of u(x)v(1 − x) = u(1 − φ(x))v(φ(x)) and φ(x) 6= 1 − x.

Given the assumptions on u and v, φ(x) is uniquely determined for each x ∈ [0, 1]. The function φ is strictly increasing and reﬂects the diﬀerence of the two players’ utility functions. The function φ associates an amount of player 1’s demand with its Nash-product equivalent amount of player 2’s

3 demand. As was shown in In (2014), the function φ provides a way to judge relative fairness of the two players’ demands in the Nash demand game. We ﬁrst state the results of In (2014), on which our results in this paper build.

∞ Theorem 1 The ﬁctitious play process (xt, yt)t=0 and its associated belief ∞ path (ft, gt)t=1 converge to a pure-strategy Nash equilibrium in the Nash demand game if and only if (x0, y0) 6∈ {(x, y)| x + y 6= 1, 0 < x < 1, 0 < y < 1, and y = φ(x)}.

In the literature, a game is said to have the ﬁctitious play property if for every ﬁctitious play process its associated belief path converges to a Nash equilibrium, possibly in mixed strategies, of the game.

Theorem 2 The Nash demand game has the ﬁctitious play property.

3 Results

3.1 Exact implementation of the Nash solution

Note that the limit to which the fictitious play process converges depends on the initial demands and therefore the fictitious play process alone cannot select a particular equilibrium out of the many Nash equilibria of the Nash demand game. That is why we now turn to the players’ simultaneous decisions on their initial demands. We consider them as a standalone one- shot game subsuming the subsequent demands in the payoff functions. That ∞ is, the sequence (xt, yt)t=1 is determined by the fictitious play and only the initial demands (x0, y0) are determined in the one-shot game, as specified below:

• The set of players: {1, 2}.

• The strategy sets for each player is [0, 1]. That is, players 1 and 2

choose their initial demands x0 ∈ [0, 1] and y0 ∈ [0, 1] respectively.

4 • The payoffs for each player are the limit of average payoffs as defined below.

T 1 X lim u¯T (x0, y0) ≡ lim u(xt)I[x +y ≤1](xt, yt) T →∞ T →∞ t t T t=0

T 1 X lim v¯T (x0, y0) ≡ lim v(yt)I[x +y ≤1](xt, yt). T →∞ T →∞ t t T t=0

For simplicity, we omit the subscript 0 and write x, y instead of x0, y0 for the rest of this section since we only deal with the initial demands. We call this one-shot game the initial demand game Γ0. We state our ﬁrst result below.

N N Theorem 3 In the initial demand game Γ0, (x , y ) is the unique Nash equilibrium.

Therefore, we can state that the “Nash” solution is the “Nash” equilibrium of the “Nash” demand game with ﬁctitious play. Theorem 3 implies that the initial demand game Γ0 (together with the subsequent ﬁctitious play) exactly implements the Nash solution. This implementation is one in complete information environments, where the utility functions u and v are common knowledge among the players, but not known to the planner.1 Now we turn to an approximate implementation of the Nash solution.

3.2 Approximate implementation of the Nash solution

Consider another one-shot game to determine the initial demands, where one-period future payoﬀ is discounted by a rate δ, where δ ∈ [0, 1).2 The one-shot game is speciﬁed below:

• The set of players: {1, 2}.

1 Serrano (2005) explained how we can view the Nash (1953) program, which relates axiomatic solutions to equilibria of non-cooperative games, in the context of the implementation theory. 2 Alternatively, (1 − δ) can be interpreted as a probability of breakdown.

5 • The strategy sets for each player is [0, 1]. That is, players 1 and 2

choose their initial demands x0 ∈ [0, 1] and y0 ∈ [0, 1] respectively.

• The payoffs for each player are the discounted average payoff of the infinite sequence of payoffs as defined below.

∞ X t u¯δ(x0, y0) ≡ (1 − δ) δ u(xt)I[xt+yt≤1](xt, yt) t=0

∞ X t v¯δ(x0, y0) ≡ (1 − δ) δ v(yt)I[xt+yt≤1](xt, yt). t=0

We call this one-shot game the initial demand game Γδ. We show an approximate implementation of the Nash solution by the initial demand game Γδ. We can approximate the Nash division by an - equilibrium defined below if we are allowed to make the players sufficiently patient (that is, δ gets sufficiently close to 1).

Definition (Radner, 1980) A strategy profile is an -equilibrium if no player has an alternative strategy that increases its payoff by more than . Furthermore, as the players make a more accurate comparison of payoffs ( → 0) and become more patient (δ → 1) accordingly, the set of -equilibria in the neighborhood of the Nash division shrinks and the only -equilibrium remaining is the Nash division.

Theorem 4 Consider the initial demand games {Γδ : δ ∈ [0, 1)}. For any open ball around (xN , yN ), there exist > 0 and δ¯ < 1 such that for any δ > δ¯ ¯ all -equilibria of Γδ are in the open ball. As → 0 and δ → 1 accordingly, ¯ the set of -equilibria of Γδ (for any δ > δ) shrinks and the only -equilibrium remaining is (xN , yN ).

In this result, the players’ estimation errors () when choosing initial demands help select particular equilibria whereas mutation or random experimentation in the evolution process has such a role in usual evolutionary or learning models.

6 4 Discussions

4.1 Sophistication levels of players

The players exhibit different levels of sophistication between their decisions in the initial demand game and those in the fictitious play. We provide two possible explanations. First, even though the players make myopic choices only considering the history of the opponents’ demands in the fictitious play, they are much more sophisticated in the initial period because of the im- portance of the initial demands that will determine the whole sequence of subsequent demands. Second, the players may delegate their decisions in the subsequent periods to their agents who are less sophisticated but good at such a simple task as keeping records of the opponent’s choices.

4.2 Generalization

We presented our results using the classic fictitious play and a particular tie-breaking rule. As the theorems of In (2014) work generally for weighted fictitious play and other tie-breaking rules as well, it can be shown that our results in this paper can also be generalized. Consider weighted fictitious play, where the relative frequencies are generalized such that earlier observations are given less weights.3 More specifically, we obtain a weighted fictitious play by replacing the relative frequencies ft(x) and gt(y) with the following ρ ρ weighted relative frequencies ft (x) and gt (y) in equations (1) and (2). For ρ ρ any ρ ∈ [0, 1], weighted relative frequencies ft (x) and gt (y) are defined recursively. For t ∈ {0, 1, 2,...} and x, y ∈ [0, 1],

 ρ ρN(t)ft (x)+1 ρ  ρN(t)+1 if player 1 chooses x in period t ft+1(x) ≡ ρ ρN(t)ft (x)  ρN(t)+1 otherwise, and

 ρ ρN(t)gt (y)+1 ρ  ρN(t)+1 if player 2 chooses y in period t gt+1(y) ≡ ρ ρN(t)gt (y)  ρN(t)+1 otherwise,

3 For general versions of the classic ﬁctitious play, see Fudenberg and Levin (1998), Camerer and Ho (1999), and Leslie and Collins (2006), for example.

7 ρ where N(0) ≡ 0, N(t + 1) ≡ ρN(t) + 1 for t ∈ {0, 1, 2,...}, and f0 (x) ≡ ρ g0(x) ≡ 0 for any x. It can be shown that our theorems continue to hold for any weighted fictitious play if the two players use the same weight ρ and ρ ∈ (0, 1]. We used a particular tie-breaking rule: ties are broken in favor of the highest demand. Suppose the two players use another tie-breaking rule such that each chooses the low demand instead of the high demand when it is indifferent between just two different demands (i.e. high and low demands). Then we can check that our theorems still hold. Suppose the two players use other tie-breaking rules such that player i chooses bi(< 1) when it is indifferent among all demands in [0, 1]. Such situations occur only if the initial demands are such that x0 = 1 or y0 = 1. We can show that the fictitious play process and its associated belief path converge to (1−yN , yN ) if N N N N (x0, y0) = (1, y ), and to (x , 1 − x ) if (x0, y0) = (x , 1) (see Proposition 1 in the appendix). Furthermore, based on Proposition 1 in the appendix, we also know that player 1’s set of best responses when player 1 expects player 2 to choose yN and player 2’s set of best responses when player 2 expects player 1 to choose xN remain intact as [0, 1] in the initial demand N N game Γ0. Therefore, (x , y ) remains as a Nash equilibrium of Γ0 and an

-equilibrium of Γδ.

APPENDIX

Proof of Theorem 3. From Lemma 2 of In (2014), we can derive the payoﬀs for each player in the initial demand game Γ0, as follows:

8   u(x) if x + y = 1 and min{x, y}= 6 0    u(1 − y) if [x + y < 1 and y > φ(x)] or   [x + y > 1 and y < φ(x)]    u(x) if [x + y < 1 and y < φ(x)] or lim u¯T (x, y) = T →∞ [x + y > 1 and y > φ(x)]   2  (u(x)) if x + y < 1, x 6= 0, y 6= 0, and y = φ(x)  u(1−y)  (u(1−y))2  if x + y > 1, x 6= 1, y 6= 1, and y = φ(x)  u(x)   0 otherwise,   v(y) if x + y = 1 and min{x, y}= 6 0    v(y) if [x + y < 1 and y > φ(x)] or   [x + y > 1 and y < φ(x)]    v(1 − x) if [x + y < 1 and y < φ(x)] or lim v¯T (x, y) = T →∞ [x + y > 1 and y > φ(x)]   2  (v(y)) if x + y < 1, x 6= 0, y 6= 0, and y = φ(x)  v(1−x)  (v(1−x))2  if x + y > 1, x 6= 1, y 6= 1, and y = φ(x)  v(y)   0 otherwise.

Note that player 1’s payoff function is continuous except at (1, 0) and (x, y) such that y = φ(x), and that player 2’s payoff function is continuous except at (0, 1) and (x, y) such that y = φ(x). From the payoff functions described above, we obtain player 1’s best response correspondence (its graph is illustrated in Figure 1)

  ∅ if y = 0   [0, φ−1(y)) ∪ [1 − y, 1] if 0 < y < yN x∗(y) = (3) N  [0, 1] if y = y   ∅ if y > yN , and player 2’s best response correspondence

  ∅ if x = 0   [0, φ(x)) ∪ [1 − x, 1] if 0 < x < xN y∗(x) = (4) N  [0, 1] if x = x   ∅ if x > xN .

We obtain the unique Nash equilibrium in the initial demand game Γ0 by

9 y 6

(xN , yN ) y = φ(x) A yN AU @ s @ @

@ @ y = φ(x) @

@ @

@ @ - 0 N 1 x c x c

Figure 1: Player 1’s Best Response Correspondence combining the two best response correspondences.

Proof of Theorem 4. In this proof, we omit the subscript 0 and write x, y instead of x0, y0 (to denote the initial demands) for simplicity. Consider a hypothetical one-shot game where we replace the payoﬀ functions of the initial demand game

Γδ by (limδ→1 u¯δ(x, y), limδ→1 v¯δ(x, y)). Since limδ→1 u¯δ(x, y) = limT →∞ u¯T (x, y) and limδ→1 v¯δ(x, y) = limT →∞ v¯T (x, y), the best response correspondences are as speciﬁed in equations (3) and (4) in section 3.1. In the hypothetical game, for any small > 0, player 1’s -best response correspondence (its graph is illustrated in Figure 2) is  −1  [u (u(1 − y) − ), 1) if y = 0   [0, φ−1(y)) ∪ [u−1(u(1 − y) − ), 1] if 0 < y < y(1) x∗(y) = −1 −1 (1) (2)  [0, 1] or [0, φ (y)) ∪ (φ (y), 1] if y ≤ y ≤ y   [u−1(u(φ−1(y)) − ), φ−1(y)) if y > y(2),

10 y 6

(xN , yN )

y = φ(x) A (2) A y N AU y y(1) s @ @

@ @

y = φ(x) @

@ @

@ @ - 0N 1 x c x c

Figure 2: Player 1’s -Best Response Correspondence where u−1 and φ−1 are inverse functions of u and φ respectively,

y(1) is the solution of φ−1(y) = u−1(u(1 − y) − ), and

y(2) is the solution of φ−1(y) = u−1(u(1 − y) + ).

The expression “[0, 1] or [0, φ−1(y)) ∪ (φ−1(y), 1]” needs explanation. This means “[0, 1] for some values of y ∈ [y(1), y(2)] (for example, yN ) and [0, φ−1(y)) ∪ (φ−1(y), 1] for the rest of y ∈ [y(1), y(2)] (for example, y(1) and y(2)). Similarly, for any small > 0, player 2’s -best response correspondence is

 −1  [v (v(1 − x) − ), 1) if x = 0   [0, φ(x)) ∪ [v−1(v(1 − x) − ), 1] if 0 < x < x(1) y∗(x) = (1) (2)  [0, 1] or [0, φ(x)) ∪ (φ(x), 1] if x ≤ x ≤ x   [v−1(v(φ(x)) − ), φ(x)) if x > x(2),

11 where v−1 and φ−1 are inverse functions of v and φ respectively,

x(1) is the solution of φ(x) = v−1(v(1 − x) − ), and

x(2) is the solution of φ(x) = v−1(v(1 − x) + ).

For any small > 0, the set of -equilibria of the hypothetical game is illustrated as a ﬁgure shaded with slanted lines in Figure 3. This set is a subset of

{x : 1 − v−1(v(y(2)) + ) ≤ x ≤ φ−1(v−1(v(y(2)) + ))}×

{y : 1 − u−1(u(x(2)) + ) ≤ y ≤ φ(u−1(u(x(2)) + ))}.

y 6

(xN , yN )

−1 (2) A φ(u (u(x ) + )) A

yN @ AU

@ @ @ s @ 1 − u−1(u(x(2)) + ) @

- 0 1 − v−1(v(y(2)) + ) φ−1(v−1(v(y(2)) + )) 1 x xN Figure 3: -Equilibria

Now consider the initial demand games {Γδ : δ ∈ [0, 1)}. If we choose a ¯ ¯ suﬃciently large δ < 1, then for any δ > δ the set of -equilibria of Γδ can be made arbitrarily close to a subset of

{x : 1 − v−1(v(y(2)) + ) ≤ x ≤ φ−1(v−1(v(y(2)) + ))}×

12 {y : 1 − u−1(u(x(2)) + ) ≤ y ≤ φ(u−1(u(x(2)) + ))}.

Note that as → 0, 1 − v−1(v(y(2)) + ) → xN ,

φ−1(v−1(v(y(2)) + )) → xN ,

1 − u−1(u(x(2)) + ) → yN , and

φ(u−1(u(x(2)) + )) → yN .

Therefore, (1) For any open ball around (xN , yN ), we can choose > 0 and δ¯ < 1 such that ¯ for any δ > δ all -equilibria of Γδ are in the open ball. (2) Any division (x, y) other than (xN , yN ) can be removed from the set of - ¯ equilibria of Γδ (for any δ > δ) if we choose a suﬃciently small > 0 and a suﬃciently large δ¯ < 1 accordingly. The only division which cannot be removed in such a way and remains as an -equilibrium is the Nash division (xN , yN ).

Proposition 1 Suppose player 1 chooses b1(6= 1) whenever it is indifferent among all demands in [0, 1] and player 2 chooses b2(6= 1) whenever it is indifferent among all demands in [0, 1]. Then the following hold: ∞ ∞ (1) The fictitious play process (xt, yt)t=0 and its associated belief path (ft, gt)t=1 N N N N converges to (1 − y , y ) if (x0, y0) = (1, y ) or (x0, y0) = (0, y ). ∞ ∞ (2) The fictitious play process (xt, yt)t=0 and its associated belief path (ft, gt)t=1 N N N N converges to (x , 1 − x ) if (x0, y0) = (x , 1) or (x0, y0) = (x , 0).

Proof. We only prove statement (1) because statement (2) can be proved similarly given the symmetry. Consider the case with the initial demands (x0, y0) = (1, yN ). We want to show the ﬁctitious play process converges to (1 − yN , yN ). N N We have (x1, y1) = (1 − y , b2), and for t ∈ {2, 3, 4,...} xt ∈ {1 − y , 1 − b2} and N N N N yt ∈ {y , b2}. If b2 = y , then (xt, yt) = (1 − y , y ) for all t ∈ {2, 3, 4,...}. We N N assume b2 6= y . Suppose b2 > y . To ﬁgure out (xt, yt) for t ∈ {2, 3, 4,...}, note that for any t ∈ {2, 3, 4,...},

 tft(1−b2) N N  t+1 if gt(y )u(1 − y ) ≥ u(1 − b2) ft+1(1 − b2) = tft(1−b2)+1 N N  t+1 if gt(y )u(1 − y ) < u(1 − b2),

13 and  N tgt(y ) t N N  t+1 if t−1 ft(1 − b2)v(b2) ≥ v(y ) gt+1(y ) = N tgt(y )+1 t N  t+1 if t−1 ft(1 − b2)v(b2) < v(y ).

N 1 N 2 We know f2(1 − b2) = 0 and g2(y ) = 2 . Therefore, g3(y ) = 3 , i.e. y2 = N 1 N N y . If 2 u(1 − y ) ≥ u(1 − b2) then f3(1 − b2) = 0, i.e. x2 = 1 − y and N N 1 N (xt, yt) = (1 − y , y ) for all subsequent periods. If 2 u(1 − y ) < u(1 − b2) then 1 1 N 2 N f3(1 − b2) = 3 , i.e. x2 = 1 − b2. If 2 u(1 − y ) < u(1 − b2) ≤ 3 u(1 − y ), 1 N N 3 N then f4(1 − b2) = 4 , i.e. x3 = 1 − y and g4(y ) = 4 , i.e. y3 = y since the 1 N N N combination of 2 u(1 − y ) < u(1 − b2) and u(1 − y )v(y ) > u(1 − b2)v(b2) 1 N N N implies 2 v(b2) < v(y ). Again, (xt, yt) = (1 − y , y ) for all subsequent periods. 2 N Consider the remaining case: 3 u(1 − y ) < u(1 − b2). We deﬁne a state variable z(t) for t ∈ {3, 4, 5,...} as follows:

t N N N • z(t) = (−−) if t−1 ft(1 − b2)v(b2) ≥ v(y ) and gt(y )u(1 − y ) ≥ u(1 − b2),

t N N N • z(t) = (+−) if t−1 ft(1 − b2)v(b2) ≥ v(y ) and gt(y )u(1 − y ) < u(1 − b2),

t N N N • z(t) = (−+) if t−1 ft(1 − b2)v(b2) < v(y ) and gt(y )u(1 − y ) ≥ u(1 − b2),

t N N N • z(t) = (++) if t−1 ft(1 − b2)v(b2) < v(y ) and gt(y )u(1 − y ) < u(1 − b2). 1 N 2 Recall f3(1 − b2) = 3 and g3(y ) = 3 . This means z(3) = (++). Since u(1 − N N y )v(y ) > u(1 − b2)v(b2), z(4) ∈ {(++), (−+), (−−)}. If z(t) = (++) for all t ∈ {3, 4,..., t¯} then z(t¯ + 1) ∈ {(++), (−+), (−−)}. Let t∗ be the smallest among t > 3 such that z(t) 6= (++). If z(t∗) = (−+), then z(t) = (−+) and N N ∗ ∗ ∗ ∗ (xt, yt) = (1 − y , y ) for all t ∈ {t , t + 1, t + 2,...}. Suppose z(t ) = (−−). t∗−2 N t∗−1 N N Then ft∗ (1−b2) = t∗ and gt∗ (y ) = t∗ . Since u(1−y )v(y ) > u(1−b2)v(b2), z(t∗ + 1) ∈ {(++), (−+), (−−)}. If z(t) = (−−) for all t ∈ {t∗, t∗ + 1,..., t˜} then z(t˜+ 1) ∈ {(++), (−+), (−−)}. Let t∗∗ be the smallest among t > t∗ such that ∗∗ N N z(t) 6= (−−). If z(t ) = (−+), then z(t) = (−+) and (xt, yt) = (1 − y , y ) for all t ∈ {t∗∗, t∗∗ + 1, t∗∗ + 2,...}. If z(t∗∗) = (++), we repeat the above argument. Eventually, the state variable gets trapped at (−+) for the following reason. Since N N t t−2 N u(1−y )v(y ) > u(1−b2)v(b2), then for a suﬃciently large t, t−1 t v(b2) < v(y ) t−1 N ∞ and t u(1 − y ) ≥ u(1 − b2). Therefore, the ﬁctitious play process (xt, yt)t=0 and ∞ N N its associated belief path (ft, gt)t=1 converges to (1 − y , y ). The argument for N the case of b2 < y is similar and we omit it. N The proof for the case with the initial demands (x0, y0) = (0, y ) is simpler. N N In this case, (x1, y1) = (1 − y , 1), and for t ∈ {2, 3, 4,...} xt = 1 − y and yt ∈

14 {yN , 1}. According to the decision rule of the ﬁctitious play, for t ∈ {2, 3, 4,...},

  1 if f (0)v(1) ≥ v(yN ) y = t t N N  y if ft(0)v(1) < v(y ).

1 N 1 Note that ft(0) = t for any t ∈ {1, 2, 3,...}. Therefore, yt = y for any t > v(yN ) . Since 0 < v(yN ) < 1, we obtain the stated result.

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