- Email: [email protected]
An implementation of the Nash bargaining solution by iterated strict dominance
Journal Pre-proof An implementation of the Nash bargaining solution by iterated strict dominance Shiran Rachmilevitch
PII: DOI: Reference:
S0165-1765(20)30011-2 https://doi.org/10.1016/j.econlet.2020.108960 ECOLET 108960
To appear in:
Economics Letters
Received date : 2 January 2020 Accepted date : 11 January 2020 Please cite this article as: S. Rachmilevitch, An implementation of the Nash bargaining solution by iterated strict dominance. Economics Letters (2020), doi: https://doi.org/10.1016/j.econlet.2020.108960. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof
pro of
An implementation of the Nash bargaining solution by iterated strict dominance Shiran Rachmilevitch∗
re-
January 2, 2020
Abstract
Anbarcı (2001) presented a modification of the “divide the dollar” game,
lP
in which equal division is obtained by iterated removal of strictly dominated strategies. I extend his mechanism to a general Nash demand game and show that it implements the Nash bargaining solution.
Key Words: Iterated strict dominance; Nash bargaining; Nash demand game.
∗
Jo
urn a
JEL Codes: C71; C78.
Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. Email:
[email protected] Web: https://sites.google.com/site/profshiranrachmilevitch/
1
Journal Pre-proof
1
Introduction
In the divide the dollar game (DD), two players simultaneously announce utility demands: player 1 demands x and player 2 demands y; if the demands are jointly
pro of
feasible—i.e., if x + y ≤ 1—each player obtains his demand; otherwise, everybody obtains zero. It is easy to see that every split of the dollar is a Nash equilibrium outcome in this game. This lead many researchers to modify the game in order to obtain a unique equilibrium outcome (see Karag¨ozo˘glu and Rachmilevitch (2018) for a recent survey of the literature). One of the modifications proposed in this literature is the following, by Anbarcı (2001). In Anbarcı´s game, called DD∗ , if x + y > 1 each player obtains a dollar-share which is equal to his demand multiplied by a factor that
re-
measures the compromise that he would like his opponent to make; for example, since by demanding x player 1 effectively offers 1 − x to player 2, but player 2 demands y > 1 − x, it is as if player 1 demands that player 2 settles for a fraction λ out of 1−x ; y
player 1’s payoff in this event is λx.
lP
what he is asking for himself, where λ =
Player 2’s payoff is defined analogously. In DD∗ , equal split of the dollar is obtained by iterative removal of strictly dominated strategies. In the present note I show that the above result extends to the more general
urn a
version of DD, the Nash demand game (Nash 1953, hereafter N DG). The rules of N DG are analogous to those of DD, the difference being that the utility possibility set need not be the unit simplex, but is allowed to be a general compact, convex and comprehensive set. The DD∗ -counterpart of N DG is denoted N DG∗ . I show that in N DG∗ , the utility allocation selected by the Nash bargaining solution (Nash 1950) is obtained by iterative removal of strictly dominated strategies.
Jo
The next section provides definitions, the results are in Section 3, and Section 4 concludes.
2
Journal Pre-proof
2
Definitions
Two players, 1 and 2, face a set of feasible utility allocations, S ⊂ R2+ . This set
is compact, convex, has a non-empty intersection with R2++ , and is comprehensive;
pro of
namely, s ∈ S and 0 ≤ s0i ≤ si for both i = 1, 2 implies s0 ∈ S. The Pareto boundary of S is described by a strictly decreasing, differentiable and concave function f ; that is, if x is a feasible payoff for player 1, then f (x) is the maximum possible payoff for player 2 given that player 1 obtains x. The set of feasible payoffs for player i is Di ; that is, D1 = [0, f −1 (0)] and D2 = [0, f (0)].
The players simultaneously announce utility demands: player 1 demands x ∈ D1 and player 2 demands y ∈ D2 . If (x, y) ∈ S the resulting payoff vector is (x, y). f (x)x y
and player 2 obtains
f −1 (y)y . x
re-
Otherwise, player 1 obtains
The logic behind these
formulas is as follows. When player 1 demands x, it is as if he offers his opponent f (x). If the demands are incompatible, this can be taken to mean that player 1
lP
asks player 2 to settle for a fraction λ out of what 2 actually wants, where λ =
f (x) . y
Applying this scaling factor to player 1’s own demand results in the above payoff. Player 2’s payoff is defined analogously. Call this game N DG∗ .
3
urn a
The Nash solution (Nash 1950) to S is N (S) ≡ argmaxs∈S s1 · s2 . Let n ≡ N (S).
Results
I start by establishing that N DG∗ is a well-defined game; namely, that the payoffs it specifies when the demands are incompatible are feasible payoffs. More specifically, given the payoff assigned to player 1,
f (x)x , y
there is “enough left” to cover the payoff
Jo
assigned to player 2:
f(
f (x)x f −1 (y)y )≥ . y x
(1)
Of course, this is equivalent to the requirement that given the payoff assigned to player 2, it is possible to give player 1 what he is supposed to receive: 3
Journal Pre-proof
f −1 (y)y f (x)x ≤ f −1 ( ). y x
pro of
In what follows, I will work with (1). Lemma 1. For every pair of infeasible demands, (x, y) ∈ D1 × D2 \ S, condition (1) holds.
Proof. Every (x, y) ∈ D1 × D2 \ S satisfies one of the following: 1. (x, n2 ) ≥ (n1 , y);
re-
2. (n1 , y) ≥ (x, n2 ); 3. (x, y) ≥ (n1 , n2 ).
Clearly, at least one of the (jointly incompatible) demands has to be weakly greater
lP
than the Nash payoff; it may be that exactly one of the demands has this property (i.e., cases 1 and 2 above) or both of them do (case 3). Suppose first that exactly one of the demands exceeds that Nash payoff; w.l.o.g, assume case 1. Specifically, fix y ≤ n2 ; I will show that (1) holds for any x ≥ n1
urn a
that satisfies (x, y) ∈ / S. First, note that these x’s are those that satisfy x > f −1 (y). The inequality x > f −1 (y) is necessary, due to demands-incompatibility. It is also
sufficient, because y ≤ n2 is equivalent to f −1 (y) ≥ n1 , hence x > f −1 (y) ⇒ x > n1 . Thus, it is enough to show that (1) holds for x > f −1 (y).
Note that (1) holds as equality when x = f −1 (y) (then (1) boils down to y = y). Thus, it is enough to prove that, given the aforementioned fixed y, the derivative of
Jo
(1)’s LHS with respect to x exceeds that of the RHS. Namely, f 0(
f (x)x f 0 (x)x + f (x) f −1 (y)y )·( )≥− . y y x2
Since the RHS of the above inequality is negative and since f is decreasing, it suffices to show that f 0 (x)x + f (x) ≤ 0, or f 0 (x) ≤ − f (x) . The latter inequality is x 4
Journal Pre-proof
equivalent to x ≥ n1 , and x ≥ n1 indeed holds, because, as we saw, it is implied by x > f −1 (y).
Now consider case 3. At least one of the demands must be strictly above the Nash
pro of
payoff; w.l.o.g, suppose that y > n2 . I will show that (1) holds for any x that satisfies x ≥ n1 and (x, y) ∈ / S. These x’s are precisely those that satisfy x ≥ n1 .
Note that (1) holds when x = n1 . To see this, note that with x = n1 (1) becomes f ( n1yn2 ) ≥
f −1 (y)y . n1
Since f −1 (y)y ≤ n1 n2 it follows that
f −1 (y)y n1
≤ n2 , hence it is
enough to show that n2 ≤ f ( n1yn2 ). This inequality is equivalent to n1 ≥
n1 n2 , y
or
y ≥ n2 . Therefore, (1) holds at x = n1 . By the argument from Case 1, the derivative of (1)’s LHS with respect to x exceeds that of the RHS.
re-
The stage is now set for the main result.
Proposition 1. N DG∗ ’s iterated strict dominance equilibrium is (n1 , n2 ).
lP
Proof. W.l.o.g, consider player 1’s decision-making.
Claim 1: A demand x < n1 is strictly dominated by x∗ = n1 . Proof of Claim 1: Consider the following cases: Case 1: y ≤ n2 . Here is it obviously better to demand x∗ rather than x, as the
urn a
former yields x∗ and the latter only yields x < x∗ . Case 2: y > n2 .
Case 2.1: n2 < y ≤ f (x). In this case the demand x yields x and the demand
x∗ = n1 yields
n1 f (n1 ) y
=
n1 n2 . y
Note that xy ≤ xf (x) < n1 n2 ; the first inequality is
by assumption on y and the second is due to the uniqueness of the Nash product’s maximizer. Therefore
n1 n2 y
> x.
Jo
Case 2.2: y > f (x). Here the demand x yields
xf (x) y
while x∗ = n1 yields
n1 n2 . y
The
latter exceeds the former, due to the uniqueness of the Nash product’s maximizer. Therefore, Claim 1 is proved. Claim 2: Once the demands x < n1 and y < n2 are removed, a demand x > n1 is strictly dominated by x∗ = n1 . 5
Journal Pre-proof
Proof of Claim 2: A demand x > yields
xf (x) y
while x∗ = n1 yields
n1 n2 . y
The latter
exceeds the former, due to the uniqueness of the Nash product’s maximizer.
Conclusion
pro of
4
I have shown that Anbarcı´s (2001) DD∗ mechanism, which is defined for the case where the utility possibility set is the unit simplex, can be extended to the case of more general utility sets. The result from DD∗ extends to the more general case: namely, a prediction of a unique outcome, which is obtained by iteratively removing strictly dominated strategies. The significance of the extension comes from the fact that in
re-
the “divide the dollar” case, an equal split need not be interpreted as the outcome of the Nash bargaining solution, because any symmetric and Paretian bargaining solution recommends this outcome in this case. The extension shows that it is really
References
lP
the Nash solution which is at work here.
295-304.
urn a
[1] Anbarcı N. (2001). Divide-the-dollar game revisited, Theory and Decision, 50,
[2] Karag¨ozo˘glu, E., and Rachmilevitch, S. (2018). Implementing egalitarianism in a class of Nash demand games, Theory and Decision, 85, 495-508. [3] Nash, J.F., (1950). The bargaining problem, Econometrica, 18, 155-162.
Jo
[4] Nash, J. F., (1953). Two person cooperative games, Econometrica, 21, 128-140.
6
Journal Pre-proof
pro of
Highlights for “An implementation of the Nash bargaining solution by iterated strict dominance” Shiran Rachmilevitch∗
re-
January 2, 2020
• I generalize Anbarcı´s (2001) DD∗ game.
• The result applies to general Nash demand games.
lP
• The Nash bargaining solution is implemented.
∗
Jo
urn a
• The solution concept is iterated strict dominance.
Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. Email:
[email protected] Web: http://econ.haifa.ac.il/∼shiranrach/
1
PII: DOI: Reference:
S0165-1765(20)30011-2 https://doi.org/10.1016/j.econlet.2020.108960 ECOLET 108960
To appear in:
Economics Letters
Received date : 2 January 2020 Accepted date : 11 January 2020 Please cite this article as: S. Rachmilevitch, An implementation of the Nash bargaining solution by iterated strict dominance. Economics Letters (2020), doi: https://doi.org/10.1016/j.econlet.2020.108960. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof
pro of
An implementation of the Nash bargaining solution by iterated strict dominance Shiran Rachmilevitch∗
re-
January 2, 2020
Abstract
Anbarcı (2001) presented a modification of the “divide the dollar” game,
lP
in which equal division is obtained by iterated removal of strictly dominated strategies. I extend his mechanism to a general Nash demand game and show that it implements the Nash bargaining solution.
Key Words: Iterated strict dominance; Nash bargaining; Nash demand game.
∗
Jo
urn a
JEL Codes: C71; C78.
Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. Email:
[email protected] Web: https://sites.google.com/site/profshiranrachmilevitch/
1
Journal Pre-proof
1
Introduction
In the divide the dollar game (DD), two players simultaneously announce utility demands: player 1 demands x and player 2 demands y; if the demands are jointly
pro of
feasible—i.e., if x + y ≤ 1—each player obtains his demand; otherwise, everybody obtains zero. It is easy to see that every split of the dollar is a Nash equilibrium outcome in this game. This lead many researchers to modify the game in order to obtain a unique equilibrium outcome (see Karag¨ozo˘glu and Rachmilevitch (2018) for a recent survey of the literature). One of the modifications proposed in this literature is the following, by Anbarcı (2001). In Anbarcı´s game, called DD∗ , if x + y > 1 each player obtains a dollar-share which is equal to his demand multiplied by a factor that
re-
measures the compromise that he would like his opponent to make; for example, since by demanding x player 1 effectively offers 1 − x to player 2, but player 2 demands y > 1 − x, it is as if player 1 demands that player 2 settles for a fraction λ out of 1−x ; y
player 1’s payoff in this event is λx.
lP
what he is asking for himself, where λ =
Player 2’s payoff is defined analogously. In DD∗ , equal split of the dollar is obtained by iterative removal of strictly dominated strategies. In the present note I show that the above result extends to the more general
urn a
version of DD, the Nash demand game (Nash 1953, hereafter N DG). The rules of N DG are analogous to those of DD, the difference being that the utility possibility set need not be the unit simplex, but is allowed to be a general compact, convex and comprehensive set. The DD∗ -counterpart of N DG is denoted N DG∗ . I show that in N DG∗ , the utility allocation selected by the Nash bargaining solution (Nash 1950) is obtained by iterative removal of strictly dominated strategies.
Jo
The next section provides definitions, the results are in Section 3, and Section 4 concludes.
2
Journal Pre-proof
2
Definitions
Two players, 1 and 2, face a set of feasible utility allocations, S ⊂ R2+ . This set
is compact, convex, has a non-empty intersection with R2++ , and is comprehensive;
pro of
namely, s ∈ S and 0 ≤ s0i ≤ si for both i = 1, 2 implies s0 ∈ S. The Pareto boundary of S is described by a strictly decreasing, differentiable and concave function f ; that is, if x is a feasible payoff for player 1, then f (x) is the maximum possible payoff for player 2 given that player 1 obtains x. The set of feasible payoffs for player i is Di ; that is, D1 = [0, f −1 (0)] and D2 = [0, f (0)].
The players simultaneously announce utility demands: player 1 demands x ∈ D1 and player 2 demands y ∈ D2 . If (x, y) ∈ S the resulting payoff vector is (x, y). f (x)x y
and player 2 obtains
f −1 (y)y . x
re-
Otherwise, player 1 obtains
The logic behind these
formulas is as follows. When player 1 demands x, it is as if he offers his opponent f (x). If the demands are incompatible, this can be taken to mean that player 1
lP
asks player 2 to settle for a fraction λ out of what 2 actually wants, where λ =
f (x) . y
Applying this scaling factor to player 1’s own demand results in the above payoff. Player 2’s payoff is defined analogously. Call this game N DG∗ .
3
urn a
The Nash solution (Nash 1950) to S is N (S) ≡ argmaxs∈S s1 · s2 . Let n ≡ N (S).
Results
I start by establishing that N DG∗ is a well-defined game; namely, that the payoffs it specifies when the demands are incompatible are feasible payoffs. More specifically, given the payoff assigned to player 1,
f (x)x , y
there is “enough left” to cover the payoff
Jo
assigned to player 2:
f(
f (x)x f −1 (y)y )≥ . y x
(1)
Of course, this is equivalent to the requirement that given the payoff assigned to player 2, it is possible to give player 1 what he is supposed to receive: 3
Journal Pre-proof
f −1 (y)y f (x)x ≤ f −1 ( ). y x
pro of
In what follows, I will work with (1). Lemma 1. For every pair of infeasible demands, (x, y) ∈ D1 × D2 \ S, condition (1) holds.
Proof. Every (x, y) ∈ D1 × D2 \ S satisfies one of the following: 1. (x, n2 ) ≥ (n1 , y);
re-
2. (n1 , y) ≥ (x, n2 ); 3. (x, y) ≥ (n1 , n2 ).
Clearly, at least one of the (jointly incompatible) demands has to be weakly greater
lP
than the Nash payoff; it may be that exactly one of the demands has this property (i.e., cases 1 and 2 above) or both of them do (case 3). Suppose first that exactly one of the demands exceeds that Nash payoff; w.l.o.g, assume case 1. Specifically, fix y ≤ n2 ; I will show that (1) holds for any x ≥ n1
urn a
that satisfies (x, y) ∈ / S. First, note that these x’s are those that satisfy x > f −1 (y). The inequality x > f −1 (y) is necessary, due to demands-incompatibility. It is also
sufficient, because y ≤ n2 is equivalent to f −1 (y) ≥ n1 , hence x > f −1 (y) ⇒ x > n1 . Thus, it is enough to show that (1) holds for x > f −1 (y).
Note that (1) holds as equality when x = f −1 (y) (then (1) boils down to y = y). Thus, it is enough to prove that, given the aforementioned fixed y, the derivative of
Jo
(1)’s LHS with respect to x exceeds that of the RHS. Namely, f 0(
f (x)x f 0 (x)x + f (x) f −1 (y)y )·( )≥− . y y x2
Since the RHS of the above inequality is negative and since f is decreasing, it suffices to show that f 0 (x)x + f (x) ≤ 0, or f 0 (x) ≤ − f (x) . The latter inequality is x 4
Journal Pre-proof
equivalent to x ≥ n1 , and x ≥ n1 indeed holds, because, as we saw, it is implied by x > f −1 (y).
Now consider case 3. At least one of the demands must be strictly above the Nash
pro of
payoff; w.l.o.g, suppose that y > n2 . I will show that (1) holds for any x that satisfies x ≥ n1 and (x, y) ∈ / S. These x’s are precisely those that satisfy x ≥ n1 .
Note that (1) holds when x = n1 . To see this, note that with x = n1 (1) becomes f ( n1yn2 ) ≥
f −1 (y)y . n1
Since f −1 (y)y ≤ n1 n2 it follows that
f −1 (y)y n1
≤ n2 , hence it is
enough to show that n2 ≤ f ( n1yn2 ). This inequality is equivalent to n1 ≥
n1 n2 , y
or
y ≥ n2 . Therefore, (1) holds at x = n1 . By the argument from Case 1, the derivative of (1)’s LHS with respect to x exceeds that of the RHS.
re-
The stage is now set for the main result.
Proposition 1. N DG∗ ’s iterated strict dominance equilibrium is (n1 , n2 ).
lP
Proof. W.l.o.g, consider player 1’s decision-making.
Claim 1: A demand x < n1 is strictly dominated by x∗ = n1 . Proof of Claim 1: Consider the following cases: Case 1: y ≤ n2 . Here is it obviously better to demand x∗ rather than x, as the
urn a
former yields x∗ and the latter only yields x < x∗ . Case 2: y > n2 .
Case 2.1: n2 < y ≤ f (x). In this case the demand x yields x and the demand
x∗ = n1 yields
n1 f (n1 ) y
=
n1 n2 . y
Note that xy ≤ xf (x) < n1 n2 ; the first inequality is
by assumption on y and the second is due to the uniqueness of the Nash product’s maximizer. Therefore
n1 n2 y
> x.
Jo
Case 2.2: y > f (x). Here the demand x yields
xf (x) y
while x∗ = n1 yields
n1 n2 . y
The
latter exceeds the former, due to the uniqueness of the Nash product’s maximizer. Therefore, Claim 1 is proved. Claim 2: Once the demands x < n1 and y < n2 are removed, a demand x > n1 is strictly dominated by x∗ = n1 . 5
Journal Pre-proof
Proof of Claim 2: A demand x > yields
xf (x) y
while x∗ = n1 yields
n1 n2 . y
The latter
exceeds the former, due to the uniqueness of the Nash product’s maximizer.
Conclusion
pro of
4
I have shown that Anbarcı´s (2001) DD∗ mechanism, which is defined for the case where the utility possibility set is the unit simplex, can be extended to the case of more general utility sets. The result from DD∗ extends to the more general case: namely, a prediction of a unique outcome, which is obtained by iteratively removing strictly dominated strategies. The significance of the extension comes from the fact that in
re-
the “divide the dollar” case, an equal split need not be interpreted as the outcome of the Nash bargaining solution, because any symmetric and Paretian bargaining solution recommends this outcome in this case. The extension shows that it is really
References
lP
the Nash solution which is at work here.
295-304.
urn a
[1] Anbarcı N. (2001). Divide-the-dollar game revisited, Theory and Decision, 50,
[2] Karag¨ozo˘glu, E., and Rachmilevitch, S. (2018). Implementing egalitarianism in a class of Nash demand games, Theory and Decision, 85, 495-508. [3] Nash, J.F., (1950). The bargaining problem, Econometrica, 18, 155-162.
Jo
[4] Nash, J. F., (1953). Two person cooperative games, Econometrica, 21, 128-140.
6
Journal Pre-proof
pro of
Highlights for “An implementation of the Nash bargaining solution by iterated strict dominance” Shiran Rachmilevitch∗
re-
January 2, 2020
• I generalize Anbarcı´s (2001) DD∗ game.
• The result applies to general Nash demand games.
lP
• The Nash bargaining solution is implemented.
∗
Jo
urn a
• The solution concept is iterated strict dominance.
Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. Email:
[email protected] Web: http://econ.haifa.ac.il/∼shiranrach/
1