Efficient Non-Cooperative Provision of Costly Positive Externalities via Conditional Commitments
Jobst Heitzig FutureLab on Game Theory and Networks of Interacting Agents, Potsdam Institute for Climate Impact Analysis, PO Box 60 12 03, 14412 Potsdam, Germany, E-mail: [email protected] September 14, 2019
Abstract We consider games where individual contributions are costly but beneficial to other players, so that contributing nothing is a dominant strategy. Considering that players may be unable to write binding agreements but may make binding unilateral commitments that are conditional on others’ actions, we study a mechanism based on condi- tional commitment functions (CCFs). If players must choose their CCFs once and simultaneously, the mechanism contributes to the Nash program since its strong (or coalition-proof) equilibria realize precisely the core outcomes of the corresponding bargaining problem. If players can communicate, the outcome can thus be expected to be Pareto-efficient. Even without communication, the core outcomes may be found by simple individual learning rules. We motivate the idea in a Cournot duopoly and a public good problem and then derive our results in a very general decision-theoretic framework and give further examples from different areas of economics. JEL classification: C72, C71, F53, D47, D43, H41. Keywords: non-cooperative game, public good, conditional com- mitment, mechanism for cooperation, core implementation, learning in games, Cournot competition, package deal
1 1 Introduction
Situations in which players’ actions have opposite effects on their own and other players’ payoffs abound in both economics and politics. In many cases, e.g., the provision of a public good, each player prefers to do “less” while all others prefer if she does “more”. If the rules of the game do neither permit binding agreements between players nor provide a possibility for punishment, e.g., if it is best modelled as a one-shot simultaneous move non-cooperative game, the usual analysis maintains that rational players would settle on quite inefficient equilibrium actions, e.g., provide no or only little amounts of the public good. We show in this paper that if the situation does not allow for binding agreements but allows players to commit themselves unilaterally to choose their actions conditional on similar commitments by other players, they can settle on an equilibrium in commitments that lead to efficient outcomes. We also show that these equilibria are both strategically very stable and are likely to come about when players apply certain plausible learning rules, even if no communication or signalling is possible and commitments are chosen non-cooperatively. The proposed mechanism is inspired by the National Popular Vote In- terstate Compact (Bennett and Bennett, 2001; Muller, 2007), which can be interpreted as a combination of unilateral but mutually dependent, condi- tional but binding commitments as follows. In 2006, California passed a state law that commits the state to allocate their share of votes in the Elec- toral College to the winner of the popular vote, i.e., the presidential candidate that received the largest share of ballot votes in the US presidential elections, under the condition that at least as many other states pass a similar law so that their joint share of votes in the Electoral College exceeds 50 percent. As of July 2019, 15 federal states of the US plus the District of Columbia have each passed such a bill, representing together only roughly 36.4 percent of the Electoral College, but in another nine states, which together would suffice to meet the threshold, such a bill is pending. Of course, announce- ments to act in certain ways given that others act in certain ways are very common in all kinds of situations in real life, but they are typically neither as precise nor as mutually conditional as this example, or if they are, as in multilateral bargaining situations, typically require an enforceable formal agreement eventually signed by all participants to make them binding. Still, there are settings in which players cannot easily sign enforceable agreements
2 but may more easily make binding unilateral commitments, such as in inter- national relations, where international treaties typically contain exit clauses but domestic law can be enforced by interest groups via supreme courts. In- deed, some authors suggest that problems such as climate mitigation and adaptation might be best addressed in this way (Coleman, 2014), and simple cases have already been studied formally (Reischmann, 2016; Reischmann and Oechssler, 2018). This motivates studying the potential for producing cooperation via mechanisms involving unilateral and mutually conditional but binding commitments more generally, between any number of players and regarding any number of issues. Before formalizing a corresponding mechanism in a very general decision- theoretic framework for a broad class of what we call costly positive external- ity problems, we introduce the main features and findings in two introductory examples, a standard Cournot oligopoly and a standard public good game. After then presenting the general framework in Sec. 2 and our main results in Secs. 3 and 4, we discuss further examples from other economic and political contexts in Sec. 5.
1.1 First introductory example: strong equilibria in a Cournot oligopoly Let us begin by considering a Cournot duopoly with two firms i = 1, 2, output quantities qi > 0, constant marginal costs of unity, and a linear inverse demand function giving the price as P (q1 + q2) = 4 − q1 − q2, so that i’s profits are ui = qi(3 − q1 − q2). The unique Nash equilibrium of the one- shot simultaneous move non-cooperative game of choosing the values qi (i.e., 0 0 0 the Cournot equilibrium) is q1 = q2 = 1 and gives ui = 1. Under collusion and joint profit maximization the players would instead choose q1 = q2 = 3/4 and get ui = 9/8 > 1. Let us now take the Cournot equilibrium as a reference point, call ai = 1 − qi the output reduction of i, and restrict ourselves to the reduction game with a1, a2 ∈ [0, 1]. Without possibilities for agreement or punishment, this game is completely boring from a classical point of view, since each player has a strictly dominant strategy, ai = 0, which is the most stable form of equilibrium imaginable. This is of course always the case when each player’s action space is linearly ordered, has some smallest element, and her payoff is strictly decreasing in her own action w.r.t. this ordering. The situation however becomes more interesting when we assume that
3 players do not choose their output (or output reduction) directly but via the following type of mechanism. Each player picks a continuous and strictly increasing conditional commitment function (CCF) ci : [0, 1] → [0, 1] with ci(0) = 0 that represents her principal willingness to reduce her output by ai = ci(a−i) conditional on her opponent’s reduction of output a−i. Also, each player may commit unilaterally to actually realize an output reduction that corresponds to the largest intersection point (˜a1, a˜2) of the curves (a1, c2(a1)) and (c1(a2), a2) under the condition that her opponent makes the same com- mitment. Then, once the CCFs c1, c2 are revealed, the point (˜a1, a˜2) can be computed and either the outputs qi = 1 − a˜i will be realized if both players have made the commitment, or otherwise q1 = q2 = 1 is realized. Fig. 1(a) shows an example of two CCFs and their outcome under this mechanism. What if both players act as “satisficers” and simply pick that CCF which corresponds to their own indifference curve through the Cournot point? We do not need to compute this CCF1 in order to see that simply by the definition of indifference curves, the resulting intersection point (˜a1, a˜2) = (1/2, 1/2) would lead to zero gains for both (Fig. 1(b), dashed lines) as compared to the Cournot point. Rational players would therefore pick CCFs that run “below” their own indifference curve through the Cournot point, such as the solid lines in Fig. 1(b,c). Now what if both players realize that the combination ? ? (a1, a2) = (1/4, 1/4) is a focal point (Schelling, 1958) among the Pareto- efficient and mutually profitable output combinations, and both pick as CCF their indifference curve through that point?2 Then their largest intersection ? ? point is just (a1, a2) so their commitments imply that they realize these ? efficient outputs ai (Fig. 1(b), solid lines). One might wonder why one should use this complicated mechanism when players could have chosen the efficient outputs right away? In that case, both players would have an incentive to deviate towards higher output. With the proposed mechanism, however, no such incentive exists: If my opponent picks as her CCF her indifference curve through some Pareto-efficient and mutually profitable point a? (e.g., thick solid line in Fig. 1b), the result will lie on that curve no matter what CCF I choose myself. Because of its Pareto-efficiency, a? is my favourite point on that curve, hence my own indifference curve
1 q 2 It would be ci(a−i) = ( a−i + 4a−i − a−i)/2. 2 But setting it to zero where it would be negative, in other words, putting ci(a−i) = q 2 max{0, ( 4a−i + 16a−i − 2 − 2a−i)/4}. From here on, we always assume indifference curves are lifted to non-negative values in this way.
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Figure 1: The conditional commitment function mechanism in a Cournot duopoly with linear demand, constant marginal costs of unity, and Cournot point qi = 1. (a) CCFs specify a firm’s output reduction ai = 1 − qi in weakly increasing dependence of the other firm’s output reduction; if both are continuous, the mechanism selects the largest inter- section point. (b) Satisficers whose CCFs are the indifference curves through the Cournot point ai = 0 do not gain (dashed lines), but players whose CCFs are the indifference curves through a point on the Pareto-efficient line (dotted line) realize that point and cannot gain from a deviation. (c) Still, for any given mutually profitable point there is a Nash equilibrium in CCFs (solid lines) which are below both firms’ indifference curves (dotted lines) that realizes that point.
5 through a? is a best response to my opponent’s choice. In other words, any pair of indifference curves through a Pareto-efficient and mutually profitable point is a Nash equilibrium in CCFs. While this is a nice result so far, unfortunately the same argument shows that the pair of indifference curves through any pointa ˜ = (˜a1, a˜2) forms a Nash equilibrium that results in the realization of this pointa ˜, as long as that point lies anywhere inside the area bounded from above by the two indifference curves and from below by the Pareto-efficient line in Fig. 1 (b). In particular, results of Nash equilibria are mutually profitable but need not be Pareto-efficient. One example of such a combination can be seen in Fig. 1(c). If we consider other CCFs which are not indifference curves, we get an even larger continuum of Nash equilibria in CCFs that lead to profitable but inefficient outcomes. Indeed, any mutually profitable point (˜a1, a˜2) is realized in some Nash equilibrium in CCFs. To see this, let ci(a−i) =a ˜i if a−i > a˜−i and let ci(a−i) otherwise be strictly below both players’ indifference curves through (˜a1, a˜2) (Fig. 1(c), solid lines). Then c1 and c2 intersect in (˜a1, a˜2) but at no larger point, and no player can gain from switching to a different CCF since an alternative CCF would intersect the other only at strictly less preferred points. Still, among the unfortunate multiplicity of Nash equilibria in CCFs, some would appear much more likely outcomes than others for a number of reasons. First, there is typically only one or a few focal points that appear “natural” for some reason (symmetry, social optimality, etc.). Second, only some will emerge as the result of some plausible learning dynamics, such as iterative best- or better-response dynamics. Third, some enjoy much stronger strategic stability properties than others. Regarding the latter, it turns out that some Nash equilibria in CCFs are strong in the sense of Aumann (1959), meaning that there is no joint deviation from which both players would gain. With only two players, these strong equilibria are simply those combinations of CCFs which give Pareto-efficient and mutually profitable outcomes, as depicted by the segment of the dotted line in Fig. 1(b) that is delimited by the two dashed lines. Nicely, our first main result will be that also with more players, all strong Nash equilibria are Pareto-efficient. The concept of strong Nash equilibrium has sometimes been criticized as being of mere intellectual interest rather than of predictive value as long as it is unclear why real players would choose a strong equilibrium rather than any other equilibrium in the first place. Fortunately, our second main result will be that some simple and plausible learning rule, if applied by all players, will always result in a strong
6 Nash equilibrium and hence in a Pareto-efficient outcome. Even better, the strong Nash equilibria of the CCF mechanism are not only Pareto-efficient but even give core outcomes (Scarf, 1967). This is a concept from cooperative game theory meant to describe those outcomes that a binding agreement might bring about if it were possible. Basically, an outcome is in the core if no group of players can produce on their own an outcome which everyone in the group prefers. Because of this result, the CCF mechanism can be seen as promoting cooperation without the need for signing binding agreements, and therefore this work can be seen as contribut- ing to the Nash program (Serrano, 2005), similar to Serrano’s core market (Serrano, 1995) and Hart’s Shapley value game (Hart and Mas-Colell, 1996). In contrast to the latter two mechanisms, the CCF mechanism even produces core outcomes in a one-shot simultaneous move game rather than in a re- peated game or iterative bargaining procedure. Another way of realizing core outcomes is via a mediator (Monderer and Tennenholtz, 2009) or via equi- libria in “programs” (Tennenholtz, 2004). A folk theorem for more general forms of conditional commitments was studied by Kalai et al. (2010). The relationship between core outcomes and strong Nash equilibria can be seen more clearly in the N-player version of the Cournot oligopoly from above. Assume N > 1 firms i, quantities qi, homogenous cost functions P C(qi) = qi, total quantity Q = i qi, inverse demand function P (Q) = 4−Q, and individual profits ui = qi(3 − Q). In that case, the Cournot point is at 0 0 0 qi = 3/(N + 1), giving Q = 3N/(N + 1), P = 4 − 3N/(N + 1), and profits 0 2 ui = 9/(N + 1) − 9N/(N + 1) . So we consider output reductions ai = P 3/(N + 1) − qi, leading to a total reduction of A = i ai = 3N/(N + 1) − Q, a price of P = A + 4 − 3N/(N + 1), and total profits of U = Q(P − 1) = (3N/(N + 1) − A)(A − 3N/(N + 1) + 3). A combination of output reductions a = (ai)i is hence Pareto-efficient iff 0 = ∂U/∂A = 6N/(N + 1) − 3 − 2A, i.e., P iff A = i ai = 3(N −1)/2(N +1), leading to a price of P = 5/2 and thus to ? total profits of U = 9/4 and individual profits of ui = 9/2(N +1)−3ai/2. For the case N = 3, this surface of Pareto-efficient reduction combinations a is shown in Fig. 2 by the outer dashed triangle. Obviously, only Pareto-efficient combinations a could make stable agreements, but not all of them. For one 0 thing, since without an agreement profits ui can be expected, a firm i would 0 2 not agree to a combination a if ui < ui , i.e., if ai > 3(N − 1)/(N + 1) . For N = 3, this boundary is at ai = 3/8 as shown by the inner dashed triangle in Fig. 2. Second, it can be argued that an agreement of all N players would probably not be signed if some subgroup G of k of the N players can sign
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Figure 2: Cournot oligopoly with 3 firms, P (q1 + q2 + q3) = 4 − q1 − q2 − q3, constant marginal costs of unity, Cournot point qi = 3/4, and output reductions ai = 3/4 − qi. The large dashed triangle is the Pareto-efficient surface (where a1 + a2 + a3 = 3/4). The small inner dashed triangle contains the mutually profitable Pareto-efficient points (where also ai 6 3/8), each of which is supported by a Nash equilibrium in CCFs. The light hexagon is the core of the cooperative version of the underlying output reduction game (where also ai + aj 6 21/32). Each point in the core is supported by a strong Nash equilibrium in CCFs. The dark surface shows a canonical equilibrium CCF for firm 3 and a chosen efficient output reduction of a = 3/4 − q = (0.3, 0.25, 0.2).
8 an alternative agreement that appears preferable to them all. If one assumes that all other players would use their Cournot outputs in that case, the P optimal joint reduction AG = i∈G of G is given by AG = 3(k − 1)/2(N + 1) since that maximizes their joint profit uG = QG(3 − QG − Q−G) = (3k/(N + ? 2 2 1) − AG)(AG + 3/(N + 1)), resulting in a value of uG = 9(k + 1) /4(N + 1) . Now a Pareto-efficient combination a gives a group G at least profits of ? 2 2 uG iff uG = 9k/2(N + 1) − 3AG/2 > 9(k + 1) /4(N + 1) , i.e., iff AG 6 3(2kN − k2 − 1)/2(N + 1)2. For N = 3 and k = 2, this boundary is at AG 6 21/32 as shown by the non-dashed border parts of the light shaded area in Fig. 2. The remaining set of reduction combinations a that fulfil all these constraints is the core of the output reduction game and equals the light shaded area in Fig. 2. Now consider a point a? in the core and assume all players specify “canon- ? ? ical” CCFs of the form ci(a−i) = ai iff aj > aj for all j 6= i and ci(a−i) = 0 elsewhere. The dark shaded planes in Fig. 2 show such a CCF for firm 3. Then these CCFs intersect only in 0 and a?, so the result is a?. If firms 1 ? and 2 then vary their CCFs, the result will either have a3 = 0, or a1 > a1 ? and a2 > a2, and in both cases one of the two firms will get at most the same profits as with a?. Using this kind of reasoning we will show that com- binations of canonical CCFs of core points always form strong equilibria. That this result and its converse hold in a very general set of situations, whether with or without possibilities of redistributing profits among players (i.e., in both transferable utility and non-transferable utility games), forms the equivalence between strong equilibria and core outcomes that is the first main result of this paper.
Short digression: difference from supply function equilibria In Cournot oligopolies, another slightly similar mechanism can also produce non-trivial and sometimes efficient Nash equilibria: the supply function mechanism of Grossmann (1981). Typically, a supply function for a supplier i of a certain commodity is a continuous and strictly increasing function Si mapping the commodity price p to the quantity qi of that commodity to be supplied by i. For N = 2 (and similarly for larger N), given supply functions S1,S2 and a continuous and strictly decreasing demand function P (q1 + q2), there is a unique solution p, q1, q2 of the three equations P (q1 + q2) = p and Si(p) = qi, and the supply function mechanism realizes these outputs and price. Klemperer and Meyer (1989) show that there is a multiplicity of Nash
9 equilibria in supply functions similar to what we demonstrated above for CCFs. One might thus suspect that the CCF mechanism is essentially equiv- alent to the supply function mechanism in this example. In fact, note that given any value for q−i, the equation qi = Si(P (qi+q−i)) has a unique solution qi. If i knows the demand function P , she can determine this solution and define a continuous function ci by putting ci(1−q−i) = 1−qi. However, since Si(P (qi + q−i)) is strictly decreasing in q−i, this ci(a−i) would be strictly de- creasing in a−i. So, this ci does not qualify as a CCF for our mechanism since we require CCFs to be weakly increasing. Conversely, given a continuous and strictly decreasing demand function Q(p) and a continuous and weakly in- creasing CCF ci, i can define a function Si by putting Si(p) = 1−ci(1−q−i(p)) where q−i(p) is the unique solution of q−i + 1 − ci(1 − q−i) = Q(p). But since q−i(p) is decreasing in p, so is Si(p), and thus Si does not make a typical supply function.3 Even if it were possible to get some kind of equivalence by allowing CCFs to be decreasing, this would violate the basic idea of a CCF, which is to commit to “contribute more” if the opponent does so as well. So although both mechanisms share the idea of picking functions that condition output on something else, they are not equivalent. More importantly, we will see that the CCF mechanism is not restricted to market situations where the externality is communicated via a price and the conditioning is w.r.t. some aggregate quantity, but can be applied in much more general situations, po- tentially involving heterogeneous agents, complex and only partially ordered action spaces, and non-classical preferences.
3Interestingly, Klemperer and Meyer (1989) allow for decreasing supply functions at first, and show that, given a concave demand function D = P −1 and a convex production 0 cost function C, if a feasible triple p, q1, q2 with p > C (qi) results from a supply function 0 0 0 equilibrium S1,S2, their slopes at p are Si(p) = D (p) + qi/[p − C (qi)], which might be negative. Indeed, one can see that if the supply function equilibrium p, qi is better than the 0 0 0 0 0 0 0 0 Cournot equilibrium p , qi for both firms, then p > p , qi < qi , p − C (qi) > p − C (qi ) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −qi P (q1 +q2), and thus Si(p) = D (p)+qi/[p−C (qi)] < D (p )+qi /(−qi P (q1 +q2)) = 0, i.e., Si is decreasing. In spite of this possibility, Klemperer and Meyer (1989) go on to prove the equilibrium property only for increasing supply functions, which may be because decreasing supply functions can lead to non-unique solutions p, qi. For increasing supply functions, Delgado and Moreno (2004) show that typically all coalition-proof (and thus all strong) supply function equilibria give the Cournot outcome, and no ordinary supply function equilibrium gives a better outcome, while we show that our CCF mechanism can sustain jointly optimal outcomes in even strong Nash equilibrium.
10 1.2 Second motivating example: learning to efficiently supply a public good A strong equilibrium would not be worth much if players have no way to find it and converge to it. In this section, we will see how with the CCF mechanism, players can use a simple learning rule to achieve this, leading even to exact convergence towards a strong equilibrium with a core outcome in finite time. Consider the public good of greenhouse gas emissions mitigation and as- sume a standard setting, slightly modified from Barrett (1994), where N countries reduce their emissions by an amount qi > 0, incurring quadratic 2 mitigation costs of qi /2γi, where γi > 0 is a cost coefficient, and experienc- ing linear benefits from avoided emissions-related damages worth βiQ, where P βi > 0 is a damage coefficient and Q = i qi is total mitigation. Then the 0 unique Nash equilibrium of the one-shot non-cooperative game is qi = βiγi, m P while joint welfare maximization would choose qi = Bγi with B = i βi. 0 This motivates studying the restricted game with actions ai = qi − qi > 0, with Nash equilibrium ai ≡ 0. Because of the restriction ai > 0, ui is de- creasing in ai and increasing in each aj, j 6= i, so the restricted game forms a typical example of a costly positive externality problem in our sense. Note that in contrast to the Cournot oligopoly, this example does not have transferable utility because the cost functions are non-linear. Hence the identification of the core is more complicated here since one needs to analyse the utility of individual group members rather than only the total utility of a group, and the inequalities that define the core are non-linear rather than linear, resulting in a curved core geometry. For N = 3, Fig. 3 shows that the core (delimited by thick blue lines), even though it is part of the Pareto-efficient manifold, also does not always contain the point of joint welfare maximization (red dot). Now assume players start with flat CCFs ci(a−i) ≡ 0, but use the follow- ing learning rule to iteratively adjust their CCFs. In each iteration, some player i (e.g., a randomly drawn one) identifies her favourite point o on the manifold defined by the other players’ current CCFs, i.e., that a for which ui(a) is largest among all a that fulfill the conditions aj 6 cj(a−j) for all j 6= i. She then replaces her current CCF by a new one, ci, which is between her canonical CCF for o and her indifference curve through o, here denoted o o ? byc ˜i . In other words, she puts ci(a−i) 6 c˜i (a−i) where aj < aj for some o ? j 6= i, and oi 6 ci(a−i) 6 c˜i (a−i) where aj > aj for all j 6= i. In this, the
11 o indifference curve is defined as follows:c ˜i (a−i) is the largest value of ai for which ui(ai, a−i) > ui(o), or zero if this holds for no ai. For example, she o could put pick a random number 0 6 λ 6 1 and put ci(a−i) = λc˜i (a−i) where ? o ? aj < aj for some j 6= i, and ci(a−i) = (1 − λ)oi + λc˜i (a−i) where aj > aj for all j 6= i. Fig. 3 shows the results of 100 different realizations of the resulting learning trajectories (colored lines), each 30 rounds long, clearly converging from a = 0 towards some point in the core (dark grey dots), and one can also see that the limit points cover the whole core. This learning rule can be interpreted as players treating others’ CCFs as a set of offers to which they respond by making a set of counteroffers that lead to mutual improvement. By offering at most their indifference curve through that point, they make sure the future outcomes can only be better than that point. If players use the even simpler rule of offering exactly their indifference curve, then the process usually converges already after one round, either landing on one of the corners of the core (if initial CCFs are all zero; blue trajectories and dots) or inside it (if using other initial CCFs; light green dots). We will show in Sec. 4 that this will in general be so, and that with the above learning rule, if the outcomes converge, then they converge to a point in the core.
2 The framework of costly positive external- ity problems
2.1 Actions
Let I be a set of N > 2 players. For each i ∈ I, let Ai be a nonempty set of possible actions for i, let 6 be a partial order on Ai representing the “magnitude” of actions, i.e., a reflexive, transitive, and antisymmetric binary 4 relation which is not necessarily linear, and assume (Ai, 6) is a complete W lattice, i.e., every subset B ⊆ Ai has a supremum (a least upper bound) B V and an infimum (a greatest lower bound) B in Ai. Denote with 0 and 1 W V the smallest and largest elements of Ai and note that ∅ = Ai = 0 and W V Ai = ∅ = 1. Each ai ∈ Ai represents a possible “contribution” of i and 0 0 ai > ai means that ai is a “larger” contribution than ai.
4 0 0 0 I.e., one may have neither ai 6 ai nor ai 6 ai for some pairs ai, ai ∈ Ai. Note that some authors call this “incomplete”, but we reserve the term “complete” for its lattice- theoretic or topological meaning.
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Figure 3: Learning strong equilibria and core outcomes in a public goods problem with three players. Starting with some CCF profile, players iteratively update their CCF to one that passes through their favourite point on all other players’ current CCFs and lies on or below their indifference curve through that point (compare Fig. 1). If players “satisfice” by using their indifference curves, they usually converge after just one round, either to a corner (blue dots, if starting from zero CCFs) of the core (thick blue lines: core boundaries, additional dashed lines: curvature on core manifold) or to some other core points (light green dots, if starting from non-zero CCFs). With CCFs properly below their indifference curves, they converge eventually (here shown: 30 rounds) to the core (colored trajectories, dark grey dots).
13 For each group of players G ⊆ I, let −G = I\G denote the complement of Q G and let AG = i∈G Ai be the set of group actions aG = (ai)i∈G prescribing an individual action ai ∈ Ai for each player in the group. We also use the notation G − H = G \ H and G − i = G \{i}. Endow AG with the product 0 0 0 partial order 6 defined by aG 6 aG for aG, aG ∈ AG iff ai 6 ai for all i ∈ G. 0 0 Denote with < the asymmetric part of 6, i.e., write aG < aG if aG 6 aG 0 and aG 6= aG, and denote with the strictly more restrictive partial order 0 0 defined by aG aG iff ai < ai for all i ∈ G. A group action aI for the whole set I is also called an action profile, we write a ∈ A = AI for short, and with a slight abuse of notation we write a = (aG, a−G) ∈ A when composing group actions aG ∈ AG and a−G ∈ A−G. In applications, the action space Ai will often be the cartesian product Q k Ai,k of a finite number of linearly ordered issue action spaces Ai,k each of which is either finite or a real interval. The partial order 6 will then typically 0 0 be either the product order, with ai 6 ai iff ai,k 6 ai,k for all issues k, or a lexicographic order, or a combination of both types. E.g., in an international climate policy context, ai,1 might be country i’s emissions mitigation, ai,2 and ai,3 might be its investments into a climate adaptation fund and research into renewable energy, and ai,4 a binary variable indicating its agreement to ban deforestation.
2.2 Preferences
For each i ∈ I, let a−i, then a
14 0 i ∈ G and a ≺i a for at least one i ∈ G. Also, let the strict partial ordering 0 ≺≺G be the intersection of the strict partial orderings ≺i, i.e., a ≺≺G a iff 0 a ≺i a for all i ∈ G. Note that ≺≺G is strictly more restrictive than ≺G. Finally, let
2.3 Unique Nash equilibrium of the base game When a CPEP is interpreted as a one-shot game in which each player i ∈ I once chooses their action ai, the individual monotonicity of preferences implies that the choice ai = 0 is the unique strictly dominant strategy. Hence, no matter in what order choices are made, whether simultaneous or not, this “base game” has a unique Nash equilibrium a ≡ 0. Note that in the case where all actions can be represented by real numbers and the orderings 4i are just the usual ordering of real numbers, the base game is a potential game in the sense of Monderer and Shapley (1996), with
15 P ordinal potential P (a) = − i∈I ai.
2.4 Pareto-efficiency and the core A group of players G ⊆ I can weakly [strictly] improve upon an action profile 0 0 0 a ∈ A iff there is some deviation profile a ∈ A with a−G = 0 so that a G a 0 [a G a]. An a ∈ A is strictly [weakly] Pareto-efficient iff I cannot weakly [strictly] improve upon it. The strict [weak] core of the CPEP (I, A, 6, 4, 4s) is the set of those a ∈ A upon which no group can weakly [strictly] improve. Note that all strict [weak] core profiles are strictly [weakly] Pareto-efficient, but that cores might be empty.5 If the base game has transferable utility, weak and strict cores are identi- cal, are known to have nice geometric properties such as convexity, closedness and boundedness, and coincide with both the α- and the β-core (Zhao, 1999) because of Individual Monotonicity.6 We do not, however, assume transfer- able utility in general here, so weak and strong cores of CPEPs can differ and may not have similarly nice geometric properties. In general, the sparser the players’ preferences, i.e., the more pairs of action profiles players are undecided about, the less likely a deviation is improving, hence the larger the weak and strong core. Consider the extreme case where costs and benefits are incomparable in the sense that for all i ∈ I, c b there is a cost ordering 4i on Ai and a benefit ordering 4i on A−i such that 0 c 0 b 0 a 0.
5We slightly deviate from common practice here in using the qualifiers “weak” and “strict” in a consistent way for both preferences, Pareto-efficiency, and core properties even though many authors omit the adjective “strict” for Pareto-efficiency and use “strong” instead of “strict” in case of cores. To avoid confusion, we reserve the adjective “strong” here for “strong Nash equilibria” where it relates to deviations by groups rather than strict preferences, which allows us to also introduce “strictly strong Nash equilibria” in a natural way. 6In the Cournot duopoly example discussed above, however, we assumed that outputs are always below the Cournot level, while the common notion of Cournot game also allows for outputs above the Cournot level, e.g. to punish other firms. This implies that the (weak and strong) core of the respective base game in our sense does not coincide with the usual notion of α- or β-core of a Cournot game.
16 3 The Conditional Commitment Function (CCF) Mechanism
3.1 CCFs, canonical CCFs, and feasibility We now formalize the idea that players can commit to act conditionally on how other players commit to act. A conditional commitment function (CCF) for i ∈ I is simply a function ci : A−i → Ai which is weakly monotonic, i.e., 0 0 ci(a−i) 6 ci(a−i) whenever a−i 6 a−i. We interpret ci(a−i) as the highest action i is agreeing to take if the others’ actions are at least a−i. For a ∈ A a 0 0 a 0 and i ∈ I, define ci (a−i) = ai if a−i > a−i and ci (a−i) = 0 otherwise, and call this i’s canonical CCF for the action profile a. If i’s preferences a 0 W 0 0 are continuous, we also definec ˜i (a−i) = {ai : a ci, iff ci(a−i) > ci(a−i) for all a−i ∈ A−i. Again, ci > ci means ci > ci 0 0 0 and ci 6= ci, while ci ci means ci(a−i) > ci(a−i) for all a−i ∈ A−i. Finally, we combine the partial orders 6 on each Ci into a partial order 6 on CG via 0 0 c 6 c iff ci 6 ci for all i ∈ G. For convenience, we also extend the relations
0 0 0 0 F (cG|aH ) = {a ∈ AG : aH = aH and ai 6 ci(a−i) for all i ∈ G}.
If H = ∅, we drop aH from our notation and write F (cG) for cG ∈ CG. The following observations motivate the usage of extreme elements of F (c) to define a CCF mechanism:
17 Lemma 1 (Best realization) For H ⊆ G ⊆ I, cG ∈ CG, a−G ∈ A−G:
1. F (cG|a−G) has a largest element, denoted b(cG|a−G).
2. This b(cG|a−G) is a weakly increasing function of cG and a−G.
3. Assume that within F (cG|a−G), society prefers larger contributions, i.e., 0 0 0 a 4s a for all a, a ∈ F (cG|a−G) with a 6 a . Then b(cG|a−G) is a so- cially optimal and thus a strictly Pareto-efficient element of F (cG|a−G), and we call it the best realization of cG given a−G. 4. If a subgroup switches to the canonical CCF of the best realization, the 0 a 0 latter does not change: If b(cG|a−G) = a, cH = cH and cG−H = cG−H 0 then still b(cG, a−G) = a.
For all proofs, see the Appendix.
3.2 The mechanism and its equilibria The CCF mechanism is now defined by the following one-shot simultaneous move non-cooperative game Γ. Each i ∈ I picks a CCF ci ∈ Ci and then the action profile b(c) of Lemma 1 is realized. Before stating the main result of our paper, we restate the relevant equi- librium concepts in our notation. We do not consider randomization of strate- gies here. A CCF profile c ∈ C is a Nash equilibrium iff no individual player has a strictly profitable deviation, i.e., iff there is no i ∈ I and c0 ∈ C with 0 0 c−i = c−i so that c i c. It is a strong Nash equilibrium (Aumann, 1959) iff no group of players has a deviation that is strictly profitable for all of them, 0 0 0 i.e., iff there is no G ⊆ I and c ∈ C with c−G = c−G so that c G c. Our main equivalence result below uses a slight refinement of the latter concept.7 A CCF profile c ∈ C is a strictly strong Nash equilibrium iff no group of players has a deviation that is weakly profitable for all of them and strictly for at least one of them, i.e., iff there is no G ⊆ I and c0 ∈ C with 0 0 c−G = c−G so that c G c. Finally, we will also characterize coalition-proof equilibria, a concept gen- erally in between Nash and strong Nash, where only self-enforcing deviations by groups are considered. As in the original definition in Bernheim et al.
7This is similar to the notion of “very strong correlated equilibrium” used in Heitzig and Simmons (2012).
18 (1987), our definition is recursive on the size of I. Let c ∈ C. For |I| = 1, c is a weakly and strictly coalition-proof Nash equilibrium iff there is no 0 0 0 c ∈ C with c I c . For |I| > 1, c is weakly [strictly] self-enforcing iff for all G ( I, cG is a weakly [strictly] coalition-proof Nash equilibrium in the modified game in which the choices of −G are already fixed to c−G. For |I| > 1, c is a weakly [strictly] coalition-proof Nash equilibrium iff it is weakly [strictly] self-enforcing and there is no weakly [strictly] self-enforcing c0 with 0 0 c I c [c I c]. Note that a strong [strictly strong] Nash equilibrium is in particular weakly [strictly] coalition-proof and a weakly [strictly] coalition-proof Nash equilibrium is in particular weakly [strictly] Pareto-efficient. Our main result essentially shows that the CCF mechanism can be used to implement exactly the core profiles of the CPEP in strong or coalition-proof Nash equilibrium.
Theorem 1 For all c ∈ C and a ∈ A:
1. If c is a Nash equilibrium, then b(c) 6≺i 0 for all i ∈ I.
a 2. If a 6≺i 0 for all i ∈ I, then c is a Nash equilibrium. 3. If c is a strong Nash equilibrium, then b(c) is in the weak core.
4. If c is a strictly strong Nash equilibrium, b(c) is in the strict core.
5. If a is in the weak core and strictly Pareto-efficient, ca is a strong Nash equilibrium, and if preferences are continuous, also c˜a is a strong Nash equilibrium.
6. If a is in the strict core, ca is a strictly strong Nash equilibrium, and if preferences are continuous, also c˜a is a strictly strong Nash equilibrium.
0 0 7. Assume that for all G ⊆ I and c ∈ C, the set D = {c ∈ C : c−G = c−G 0 0 and c G c [c G c]} has a strictly G-optimal element, i.e., some 0 00 0 00 c ∈ D with c 6 G c for all c ∈ D(G). Then if c is a weakly [strictly] coalition-proof Nash equilibrium, it is already a strong [strictly strong] Nash equilibrium.
19 4 Finding core allocations by learning
4.1 Finding the corners quickly by satisficing If all players are content with a solution at least as good as a ≡ 0, they can find and settle on a corner of the core (which will then typically be strictly preferred to a ≡ 0 by all but one player after all), via a very simple learning procedure that might be termed “satisficing learning”. The idea is that players specify their CCFs not simultaneously but in a certain order, and each player specifies as her CCF her indifference curve through an action profile that is feasible w.r.t. the earlier players’ CCFs, is zero for all later players, is optimal for her among those profiles, and is maximal among those optimal profiles. Our first result in this section is that this form of best response dynamics results in a core action profile. For convenience, we will use the notations “
• We say that actions are dense iff for all i ∈ I and a, s ∈ A with a i s, 0 0 0 there is ai ∈ Ai with ai > ai and (ai, a−i) ai, we have (ai, a−i) k a and 0 (ai, a−i) ≺i a.
Theorem 2 Assume preferences are continuous and there are action profiles o1, . . . , oN ∈ A, and a CCF profile c ∈ C such that for all i ∈ I, the following hold:
i i i (1) o is an i-optimal c
i oi (2) ci is i’s indifference curve through o , i.e., ci =c ˜i . Then b(c) is in the weak core. If, in addition, actions are dense and prefer- ences strictly monotonic, b(c) is also in the strict core.
As can be seen in Fig. 3 (blue dots), the results of this learning rule for the N! possible player orderings are the corners of the core.
20 4.2 Exploring the interior by using nonzero initial pro- files We next show that a similar learning rule also leads to a core result when players already have some nonzero CCF profile c0 and then one by one replace their CCFs by their indifference curves through maximal optimal points on the other players’ current CCFs, as long as the last chosen optimal point exceeds those of all players with nonzero resulting actions.
Theorem 3 Assume preferences are continuous and there are action profiles o1, . . . , oN ∈ A, and two CCF profiles c0, c ∈ C such that for all i ∈ I, the following hold:
i 0 i (i) (1) o is an i-optimal (c
i oi (2) as in Theorem 2, ci is i’s indifference curve through o , i.e., ci =c ˜i .
N i (3) either b(c)i > 0 or o > o . Then b(c) is in the weak core. If, in addition, actions are dense and prefer- ences strictly monotonic, b(c) is also in the strict core.
As can be seen in Fig. 3 (black dots), the results of this learning rule may lie on the boundary or in the interior of the core, and rarely outside the core with some zero actions (if condition (3) is violated).
4.3 Convergence under less yielding learning rules While the satisficing behaviour studied in Theorems 2 and 3 leads quickly into the core, it appears not utterly realistic since a player early in the line will later realize she could have gotten more than provided by the optimal feasible profile at the time she made her choice if she had offered to the later players less than what was given by her indifference curve. We hence study last a more plausible, repeated version of the above learning rule in which players update their CCFs in random order and use CCFs that lie somewhere between their canonical CCF and indifference curve through the currently optimal feasible point. To still keep the rule as simple as possible, we as- sume they use random convex combinations of these two extreme choices,
21 sometimes demanding more at the risk of delaying convergence, sometimes demanding less to speed up the process. Although we are not able to prove convergence here, we can at least show that if the process converges, the limit must be in the core. For this last result, we introduce two even stronger properties of actions and preferences:
• We say that actions are metric vectors iff each Ai is a real vector space Q and d is a vector space metric on the product space A = i∈I Ai. • We say that preferences are metrically continuous and bounded iff all i ∈ I have a continuous von-Neumann–Morgenstern utility function ui :(A, d) → [0, 1].
Theorem 4 Assume actions are metric vectors, preferences are metrically t continuous and bounded, and players change their CCFs ci over discrete time t = 0, 1,... as follows:
0 (0) At t = 0, all ci ≡ 0. (1) At t > 0, some j = jt ∈ I is chosen independently according to some fixed positive probabilities pj > 0. (2) j picks some optimal currently feasible action profile o = ot ∈ t−1 arg max{uj(a): a ∈ F (c−j )} and draws some random coefficient ` = `t ∈ [0, 1] according to some fixed positive probability density func- tion φ(`) > 0.
(3) j changes her CCF to the convex combination between her canonical t o o and indifference CCFs through o, cj = `cj + (1 − `)˜cj , while all other t t−1 CCFs remain unchanged, c−j = c−j . Then almost surely the solutions bt = b(ct) do not converge for t → ∞ to some a that is not in the weak core. In particular, if they converge with positive probability, then almost surely to some a in the weak core.
Fig. 3 (paths ending in gray dots) also shows a sample of realizations of this rule, all of which apparently converge to the core. We end this section with two conjectures regarding this iterative learning procedure. Conjecture 1. Theorem 4 also holds for “strict core”. Conjecture 2. Under (basically) the assumptions of Theorem 4, bt con- verges almost surely (and hence to some element of the core).
22 5 Examples
In this final section, we illustrate the breadth of potential applications of the CCF mechanism with examples from different branches of economics.
5.1 Cournot oligopoly with concave utility We start by revisiting a more general version of Cournot oligopoly, in which P N firms choose output quantities qi > 0, leading to total supply Q = i qi 0 and a price P (Q) with P (Q) 6 0. Assuming that cost functions Ci(qi) > 0 0 are strictly convex and P (Q)qi + P (Q) is weakly decreasing in each qi, i.e., 00 0 0 that P (Q)Q 6 −2P (Q) for all Q, there is a unique equilibrium qi , and for 0 all smaller output vectors q 6 q , profits ui = P (Q)qi − Ci(qi) are strictly 0 decreasing in own output reduction ai = qi − qi > 0 and weakly increasing in output reductions aj, j 6= i: 0 0 ∂ui/∂ai = −∂ui/∂qi = Ci(qi) − P (Q)qi − P (Q) < 0, (1) 0 ∂ui/∂aj = −∂ui/∂qj = −P (Q)qi > 0. (2)
Hence the problem of choosing output reductions ai w.r.t. the Cournot level 0 qi forms a CPEP and our results apply. This shows that the CCF mechanism, in contrast to the supply function mechanism, would allow firms to find a Pareto-efficient profit vector if the CPEP has a non-empty core. Although our assumptions imply that all profit functions are continuous and concave, most core nonemptiness results from the literature do not di- rectly apply here since they assume firms may raise output beyond q0 as a reaction to a group’s deviation. However, our constraint that they don’t can be interpreted as a capacity constraint in the terminology of Lardon (2012), so that their Thm. 4.2 implies that the γ-core of the transferable-utility ver- sion of the output reduction game is nonempty. Since the latter is contained in our notion of core in the non-transferable-utility output reduction game, we can finally conclude that the CCF mechanism has strong equilibria here and all these lead to Pareto-efficient profit vectors.
5.2 Bilateral trade depending on a broker, and politi- cal package deals While the oligopoly example has one-dimensional continuous action sets, this simple example has discrete action sets, one of which is two-dimensional.
23 A broker B can either make contact, aB = 1, costing her > 0, or not make contact, aB = 0, between a firm F and a potential customer C to enable a bilateral trade transaction between them. The customer can either pay, aC = 1, or not pay, aC = 0, a fixed price p > 0 for some good she values at v > p. The firm can either pay, aF,P = 1, or not pay, aF,P = 0, a fixed premium r ∈ (, p) to the broker, and can either deliver, aF,D = 1, or not deliver, aF,D = 0, the good which she values at c ∈ [0, p−r). We assume that in reality, payment and delivery can only occur if contact was made, but to fit our framework of CPEPs, action sets must be independent. We solve this issue by assuming formally that payment and delivery can also occur without contact being made, but that they reach their targets only when contact was made and are otherwise lost. Their utilities are then
uB = aF,P r − aB, (3)
uF = aBaC p − aF,P r − aF,Dc, (4)
uC = aBaF,Dv − aC p, (5) which can easily be seen to fulfil the monotonicity requirements of a CPEP. The weak and strict core both consist of the unique action profile in which all four possible acts are performed, giving payoffs of (r − , p − r − c, v − p) > 0. Indeed, if only B and F cooperate, C won’t pay and they can get at most 0 < p − c − , if only B and C cooperate, F won’t pay or deliver and they can get at most 0 < r + v − − p, and if only F and C cooperate, B won’t make contact and they can get at most 0 < v − r − c. There are several strictly strong equilibria in CCFs that realize the core payoffs. A natural but non-minimal equilibrium set of conditions is that B conditions on the premium payment, C on making contact and delivery, and F both his acts on making contact and getting paid, which makes seven conditions. Indeed, in equilibrium B must condition on getting the premium, since otherwise F and C can jointly deviate by not offering to pay the premium, which would be a strict gain to F and no loss to C. F must condition paying the premium payment either directly or indirectly (via C’s CCF) on B making contact, since otherwise B can unilaterally deviate by not offering to make contact and still get the premium, which would be a strict gain. Likewise, either B must condition making contact or F must condition delivery directly or indirectly on C paying, since otherwise C can unilaterally deviate by not offering to pay and still get the good, which would be a strict gain. And C must condition directly or indirectly on delivery. But it suffices if B
24 conditions making contact on getting the premium, F conditions premium payment on C paying but offers delivery unconditionally, and C conditions payment on getting contact and delivery, which are just four conditions. This indicates that in situations where actions correspond to combina- tions of possible acts, a core outcome that requires a certain combination of acts may not only be realized by canonical CCFs in which each player’s combination of required acts is conditioned on all other players’ required acts, but may often be realized by simpler CCFs that condition only some of the required acts on some other required acts, using much fewer condi- tions. Especially in the context of international relations this may make an application of the CCF mechanism easier. For example, the package deal agreed on the G7 summit in Bonn 1978 contained amongst others the following significant concessions (Putnam and Henning, 1986). (A1) The US would reduce oil imports and decontrol the oil price.
(A2) The US would reduce inflation and currency fluctuations.
(B) The UK would cut tariffs in the Tokyo GATT round.
(C) Canada would increase economic growth.
(F) France UK would cut tariffs in the Tokyo GATT round.
(G) Germany would increase economic growth.
(J1) Japan would increase economic growth.
(J2) Japan would decrease their trade surplus. From the records, it appears that during negotiations, many bilateral de- mands were made, including the following 22 (Putnam and Henning, 1986). The US demanded (F), (G), and (J2). The UK demanded (A2), (G), (J1), and (J2). Canada demanded (A1) and (A2). France demanded (A1), (C), (G), (J1), and (J2). Germany demanded (A1), (A2), (B), (F), and (J1). Japan demanded (A1), (A2), and (C). With the CCF mechanism, the exact same result would have been ob- tained with the following subset of just eight conditions: The US commits to (A1) under condition (F) and to (A2) unconditionally, France commits to (F) if (G), Germany to (G) if (B), the UK to (B) if both (J1) and (J2),
25 Japan to (J1) if (C) and to (J2) unconditionally, and Canada to (C) if (A1) and (A2). Although we cannot perform a formal analysis of whether this would have formed a strong Nash equilibrium without speculating about the players’ exact preference relations, whether they are linear or not, it seems plausible that at least these CCFs would likely have formed a Nash equi- librium since any unilateral deviation would have implied the removal of all concessions except (A2) and (J2).
5.3 A supply chain In this three-player example, the action sets are continuous but not all one- dimensional, and we illustrate how the CCF mechanism may be a helpful device in recurring situations governed by uncertainty. In each time period t, a manufacturer M buys an amount x > 0 of raw materials from a supplier S against a total payment of z > 0 (corresponding to a price z/x > 0), then produces out of x an amount y ∈ [0, x] of consump- tion goods, incurring unit labour costs normalized to unity, and finally sells the amount y to a retailer R against a total payment of r > 0 (corresponding to a price r/y > 0). In period t, S can deliver at most a varying amount s(t) ∈ [s, s] and has varying unit costs d(t) ∈ [d, d], while the retailer can sell any amount at a time-varying market price p(t) ∈ [p, p]. All of s(t), d(t), p(t) and their exact distribution are previously unknown to the players, but their bounds are publicly known and the actual values are known at the beginning of each period. We assume p > d + 1 so that profits are possible even under the worst-case parameter values. Given the control variables x, y, z, r and the parameters s, d, p, profits are
uS = z − dx, uM = r − y − z, uR = py − r, U = (p − 1)y − dx, (6) and the maximal possible (and thus Pareto-efficient) total profit is
U ?(s, d, p) = (p − d − 1)s. (7)
As a benchmark situation, let us assume that M negotiates long-term contracts with both S and R independently, the negotiators deal with the uncertain parameters by only anticipating the guaranteed surplusses they can get if s = s, d = d, p = p, that in both negotiations the result of the other negotiation is anticipated correctly (as a form of rational expectations),
26 and the result is given by the Nash bargaining solution, which in this case leads to a symmetric sharing of surplusses. Then this leads to the following time-independent agreement:
x˜ =y ˜ = s, z˜ = (p + 2d − 1)s/3, r˜ = (2p + d + 1)s/3, (8) so that the prices are
z/˜ x˜ = (p + 2d − 1)/3, r/˜ y˜ = (2p + d + 1)/3. (9)
The resulting profits partially vary over time, depending on d for S and on p for R, and are inefficient since usually the supply is underutilized (˜y < s(t)):
u˜S(d) = (p + 2d − 3d − 1)s/3, (10)
u˜M = (p − d − 1)s/3, (11)
u˜R(p) = (3p − 2p − d − 1)s/3, (12) ˜ ? U(d, p) = (p − d − 1)s 6 (p − d − 1)s = U (d, p, s). (13)
Note that uM 6 uS and uM 6 uR, with equality only if d = d and p = p, hence M is generally disadvantaged in this benchmark situation. Alternatively, assume that quantities and payments are determined in each period by applying the conditional commitment function (CCF) mecha- nism with M specifying a time-independent CCF upfront and S,R specifying new CCFs each period depending on s(t), d(t), r(t). For example, M could offer S the same price as in Eqn. (9), which isz/ ˜ x˜, but conditioning also on r to make sure to stay within her budget:
z˜ y˜ c (x, r) = min x, r − r . (14) z x˜ r˜
The second term is a safe lower bound for her budget if she offers R the same price as in Eqn. (9), since her labour costs that must be subtracted from r y˜ are y, which is then at most r˜r. Similarly, when offering R this price, she also conditions on x to make sure to be able to deliver the amount y:
y˜ c (x, r) = min r, x . (15) y r˜
27 Then, in each period t, S might specify a “satisficer’s” CCF that only guar- antees her at least the same profit as in the benchmark,u ˜S, which she can achieve by using her indifference curve through the point (˜z, x˜),
z − u˜ (d(t)) c (z) = max 0, min s(t), S . (16) x,t d(t)
If R also uses her indifference curve through the point (˜y, r˜) as her CCF, which is
cr,t(y) = max{0, p(t)y − u˜R(p(t))}, (17) then the solution of the CCF mechanism results in these quantities and Pareto-efficient profits: z˜ x(t) = y(t) = s(t), z(t) = s(t), r(t) = p(t)s(t) − u˜ (p(t)), (18) x˜ R
? uS(t) =u ˜S(t), uM (t) = U (t) − u˜S(t) − u˜R(t), uR(t) =u ˜R(t). (19)
So, in this scenario, M carries the risk of fluctuating supply, costs and prices, having sometimes smaller and sometimes larger profits than S,R. Since in each period, the solution is Pareto-efficient and no single player or pair of players can generate a positive profit on their own, it is in the core of the underlying period game, so our results imply that the above combination of CCFs is a strong equilibrium in each period. This means no single player or pair of players has an incentive to specify different CCFs if they do not consider intertemporal strategic effects. (An intertemporal analysis of the resulting repeated game is beyond our scope here.) Of course, players could achieve the same result by signing two individual contracts upfront, one between M and S, including a clause that conditions their liabilities on the actions (r) of a third-party (R), and one between M and R, also including a clause that conditions their liabilities on the actions (x) of a third-party (S). However, this would introduce a circular depen- dency between several contracts which might give the firms’ legal depart- ments a headache that we believe would be larger than that caused by the mathematically much less ambiguous CCF mechanism proposed here.
28 5.4 A simple commodity exchange This example demonstrates that efficient conditioning may not require much information. We consider a simple commodity exchange for a certain good, with N = n + m + 1 players, n buyers Bi, i = 1 . . . n, m suppliers Sk, k = n + 1 . . . n + m, and a central exchange E. Bi’s possible actions are to make some monetary payment pi to E, while Sk may supply any quantity qk of the good to E. E may deliver any quantity di of the good to Bi and transfer some monetary amount tk to Sk, all these being nonnegative. Bi’s utility ui(di, pi) is continuously differentiable, strictly concave and increasing in di and strictly concave and decreasing in pi, while Sk’s utility uk(tk, qk) is continuously differentiable, strictly concave and increasing in tk and strictly concave and decreasing in qk. E only cares about meeting the budget con- P P straints, so we assume she has utility u0 = 0 iff P = pi > tk = T P P and Q = qk > di = D and otherwise u0 = −1. It is clear that un- der these assumptions, there is a unique market price γ? so that everyone’s price-taking utility optimization would lead to Pareto-efficient trade with ? ? ? ? ? ? ? ? ? ? pi = γ di , ti = γ qi , P = T , and D = Q , forming the competitive equilibrium that is the unique element of the core of this economy. We now show that with the CCF mechanism, this competitive equilibrium can be brought about by a strong equilibrium in CCFs that condition only on very few quantities and require the players to know only the market price and their own utility functions. In particular, Bi need only use a step-function CCF that depends only on the amount delivered to her, while Sk need only use a step-function CCF that depends only on the transfer she receives. E only uses a CCF where she conditions each di on the corresponding pi and each tk on the corresponding qk, and on the aggregates P and Q. More ? ? precisely, Bi puts ci(di) = pi if di > di and ci(di) = 0 otherwise, Sk puts ? ? ck(tk) = qk if tk > tk and ck(tk) = 0 otherwise. E may then use the CCF ? c0(p, q) = (d, t) = (Qp/P, P q/Q) that does not even depend on γ . It is easy to see that the resulting allocation is the above core allocation, so Theorem 1 implies that the corresponding canonical CCFs ca would form a strong equilibrium. To show that the simpler CCFs c specified here do so as well, let us assume some group G of players has an incentive to jointly specify different CCFs. If E/∈ G, the resulting action profile will meet E’s budget constraints by definition of c0. If E ∈ G, it must do so as well to be a weak improvement for E. Hence the resulting action profile will represent a trade between the players G \{E} and potentially some players outside G.
29 Let H be the set of all players outside G but trading nonzero amounts. Since each i ∈ H has specified a CCF that ensures her at least her equilibrium utility in this case, the resulting trade is a trade restricted to G ∪ H that is a weak Pareto-improvement for G ∪ H. But this is impossible since the original trade was already in the core of the economy.
5.5 A mixed exchange economy with other-regarding preferences Although in many cases, the core of an economy consists basically of its (approximate) competitive equilibria, there are several reasons why the core may be larger. One such reason would be the restriction that trade can only occur along the links of some underlying network, which restricts the set of coalitions with outside options to connected subsets of the network and thus widens the core. A different reason for a nontrivial core can be other-regarding preferences, which are featured in this example. We consider an exchange economy with three goods 1, 2, 3 that consists of a continuum of traders who are divided into three groups (“oceans”) A, B, C with different endowments, the first of which later has other-regarding preferences concerning the third group. We will see that in this case, the applicability of our results depends on the exact form of those preferences defining their monotonicity properties. Each group A, B, C is represented by a copy of the unit interval [0, 1] and consists of identical market participants i. Let us assume at first that all are i i i i i i i 1/3 sharing the same Cobb–Douglas utility function u (x1, x2, x3) = 3(x1x2x3) i i i i if x1, x2, x3 > 0 and u = −∞ otherwise, but have different initial endowment i i i i i densities e = (e1, e2, e3) = (1, 0, 1) for i ∈ A, e = (0, 1, 1) for i ∈ B, and ei = (0, 0, 1) for i ∈ C. One can see that then the unique competitive equilibrium has prices p ∝ (1, 1, 1/3), utilities ui = siU ?, and allocation densities xi = si × (1, 1, 3), where U ? = 34/3, si = 4/9 for i ∈ A ∪ B and si = 1/9 for i ∈ C. Note that by Aumann (1964), the core of the market (and thus of the corresponding “delivery” game in which players can deliver arbitrary non-negative quantities of the goods to other players) so far consists of this competitive equilibrium alone. Now assume players in A have other-regarding preferences concerning the welfare of the disadvantaged players in C. More precisely, assume that instead of ui, all i ∈ A have actual preferences based on the utility function
30 i i i i ? i ? R j i ? uˆ given byu ˆ = u if u < U /3 butu ˆ = U /3 + C dj u if u > U /3. In other words, i ∈ A only cares for C’s average utility rather than her own whenever her own utility exceeds the egalitarian utility share U ?/3. What is a price-taking-based market equilibrium under these alternative preferences? Let us assume that players can not only trade at market prices p but can also donate goods to specific other players, and note that only i ∈ A may have an incentive to make donations, and only to some j ∈ C. Hence players i ∈ B will still exploit her budget and have p · xi = p · ei. Since i ∈ A would have incentives to donate more as long as ui > U ?/3, an equilibrium i ? i ? must have u 6 U /3. If, on the other hand, u < U /3, she will not donate but trade as usual to increase ui if possible, so she will then exploit her budget and have p · xi = p · ei. Only if ui = U ?/3, she may have donated, in i i i i which case p·x 6 p·e . If no i ∈ A donates, also all j ∈ C have p·x = p·e , i i otherwise p · x > p · e . All in all, it can be shown that the resulting unique equilibrium has the same prices as before, p ∝ (1, 1, 1/3), and gives i ∈ B the same utility as before, ui = 4U ?/9. The only difference is that now all i ∈ A, while trading the same quantities as before, also donate a share of their allocation uniformly to all j ∈ C so that eventuallyu ˆi = ui = U ?/3 and uj = 2U ?/9. The corresponding final allocations are
xi = (1/3, 1/3, 1) for i ∈ A, xi = (4/9, 4/9, 4/3) for i ∈ B, xj = (2/9, 2/9, 2/3) for j ∈ C. (20)
Interestingly, the introduction of other-regarding preferences also enlarges the core, which now contains allocations that differ from the price-taking- based market equilibrium. In particular, the completely homogenous allo- cation xi ≡ (1/3, 1/3, 1) for all i is now in the core, resulting in utilities uˆi = 2U ?/3 for i ∈ A and ui = U ?/3 for i ∈ B ∪ C. This is because (i) no coalition that does not intersect A or B can generate non-zero utility because they own nothing of good 1 or 2, (ii) a coalition that does intersect A and B could only improve every member’s utility by increasing all of C’s average utility beyond its current level of U ?/3, but this would not leave enough to also increase its B-members’ utility. Indeed, assume a coalition G of size a in A, b in B, and c in C could improve all their utilities by suitable internal trade. Since the original allocation is homogenous, we can assume w.l.o.g. that in each subgroup, the resulting allocation is homogenous, and use the abbreviationsu ˆA, uB, uC for utilities and xA, xB, xC for densities inside the
31 coalition. Also let xD be the density donated to each i ∈ C \ G. Then the market clearance constraint is
axA + bxB + cxC + (1 − c)xD = aeA + beB + ceC = (a, b, a + b + c), (21) and the utility constraints are
? C C C C 1/3 U /3 < u = 3(x1 x2 x3 ) , (22) ? B B B B 1/3 U /3 < u = 3(x1 x2 x3 ) , (23) ? A A A A 1/3 U /3 = u = 3(x1 x2 x3 ) , (24) U ?/3 < uˆA − U ?/3 (25) C C C 1/3 D D D 1/3 = 3c[x1 x2 x3 ] + 3(1 − c)[x1 x2 (1 + x3 )] . (26)
One can show that this system of constraints has no solution, so the homoge- nous allocation xi ≡ (1/3, 1/3, 1) is in the core. We will now show that this homogenous allocation can be brought about by our CCF mechanism in strong equilibrium if players have slightly modified other-regarding preferences that ensure the applicability of Theorem 1. To be able to apply it, we need preferences which are monotonically decreasing in own actions and increasing in others’. In the delivery game, an action is the delivery of goods, so in its current formulationu ˆA does not have the required monotonicity properties since it increases in deliveries from A to C and decreases in deliveries from C to A or B. This can be fixed by considering a different version of other-regarding preferencesu ˜ in terms of deliveries which in our example however leads to the same behaviour asu ˆ i→j does. To this end, let y > 0 be the amount of goods delivered by i to j, i i P j→i i→j so that x = e + j(y − y ) and
Z 3 !1/3 i ? Y j X `→j j→` uˆ = U /3 + 3 dj ek + (yk − yk ) (27) C k=1 `
i ? for i ∈ A if u > U /3. To defineu ˜, we now simply drop all terms from the i→j j→i sum that have the wrong monotonicity, i.e., the terms yk − yk , giving
Z 3 !1/3 i ? Y j X `→j j→` u˜ = U /3 + 3 dj ek + (yk − yk ) (28) C k=1 `6=i
32 i ? i i for i ∈ A if u > U /3, andu ˜ = u otherwise. It is easy to see thatu ˜ now fulfills the monotonicity requirements of Theorem 1. What effect does the change fromu ˆ tou ˜ have on the price-taking-based market equilibrium? Since withu ˜, direct donations from i ∈ A to C do not increase i’s utility, i does no longer have an incentive to make donations when ui > U ?/3, so now an equilibrium may have ui > U ?/3. As a consequence, while the allocation from Eq. (20) is still an equilibrium, there are now many i ? j ? other equilibria, in all of which still u = 4U /9 for all i ∈ B, u 6 2U /9 for ? i ? all j ∈ C, and U /3 6 u˜ 6 2U /3 for all i ∈ A. None of these additional equilibria is Pareto-efficient since in them, i ∈ A retain goods that do not increase their utility but could increase others’ utility. The core of the delivery game withu ˜ is however the same as withu ˆ since a blocking coalition that can generate nonzero utility must intersect B, hence transfers from A to C can be routed via B and do not require direct donations from A to C. In particular, the homogenous allocation xi ≡ (1/3, 1/3, 1) is still in the core of the delivery game withu ˜ and can thus be realized in strong equilibrium via a suitable combination of CCFs. One such combination is the following. Put
A→B B→C B→C B→A y1 = 2/3, y1 = y2 = y2 = 1/3 (29)
X→Y and all other yj = 0. Now each player in X with X ∈ {A, B, C} commits X→Y to delivering yj density units of good j ∈ {1, 2, 3} to each player in Y ∈ {A, B, C} iff all other players do their respective deliveries, and else delivers nothing to anybody. While it may seem rather pathological to consider preferences of the kind ofu ˜i, there is a natural interpretation for them. The primary interest of the agent is to satisfy their basic needs, here assumed to be represented by the sufficiency utility level U ?/3. Once her basic needs are satisfied, her main interest is the welfare of a certain disadvantaged group of agents, here C, and she is willing to sacrifice additional own consumption for their benefit. She however prefers such a social allocation to come about not by having to donate to C directly herself, and when it comes to direct interaction with a member j of C, she is indifferent between donating to j and taking advantage of trading with j. Such preferences may also help explaining many real-world people’s apparent willingness to vote for redistribution policies but not to make significant direct donations.
33 Appendix W Proof of Lemma 1. 1. Let aG = b(cG, a−G) = F (cG, a−G). We have to show that aG ∈ F (cG, a−G), i.e., that ai 6 ci(aG−i, a−G) for all i ∈ G:
_ 0 0 _ 0 0 ai = {ai : aG ∈ F (cG, a−G)} 6 {ci(aG−i, a−G): aG ∈ F (cG, a−G)} _ 0 6 {ci(aG−i, a−G): aG ∈ F (c)} = ci(aG−i, a−G), where the last inequality follows from the monotonicity of ci and the last equality follows from F (cG, a−G) 6= ∅ because trivially 0 ∈ F (cG, a−G). 0 0 0 0 2. If cG 6 cG and a−G 6 a−G, we have F (cG, a−G) ⊆ F (cG, a−G) and W W 0 0 0 0 hence b(cG, a−G) = F (cG, a−G) 6 F (cG, a−G) = b(cG, a−G). 3. If aG ∈ F (cG, a−G) then b(cG, a−G) > aG by 1. and thus (b(cG, a−G), a−G)
Proof of Theorem 1. Note that because of the possible lack of totality in the involved orders, we have to be careful to distinguish 6 from > and 6≺ from <. 0 0 1. Assume b(c) ≺i 0 for some i. Put ci(a−i) = 0 for all a−i, and c−i = c−i. 0 0 0 Then b(c )i = 0 and b(c )−i > 0, hence b(c ) ai (i.e., 0 0 i offers at least as much as before) then a ∈ F (c ) and thus b(c ) > a by 0 a 0 definition of F and b; but b(c )−i 6 a−i by definition of c , thus b(c ) 4i a 0 by Individual Monotonicity, which is a contradiction to b(c ) i a. Hence 0 ci(a−i) >6 ai (i.e., i does not fulfill the conditions of the others) and thus 0 a 0 b(c )−i = 0 by definition of c . So a ≺i b(c ) 4i 0 by Individual Monotonicity, a a contradiction to a 6≺i 0. Hence c is a Nash equilibrium after all. 3. Assume b(c) is not in the weak core, so we can pick G and a0 with 0 0 0 a−G = 0 and a G b(c). We can now construct a deviation c for G 0 0 0 0 0 that makes b(c )G = aG: For each i ∈ G, put ci(a−i) = ai if aG−i > aG−i 0 0 0 0 and ci(a−i) = 0 otherwise, and put c−G = c−G. Then b(c )G = aG and 0 0 0 0 b(c )−G > 0 = a−G, hence b(c ) 34 4. Similarly, assume b(c) is not in the strict core and pick G and a0 with 0 0 0 0 0 a−G = 0 and now a G b(c). Define c as in 3. Then b(c ) 35 For the “strict” case, use the same proof but replace every occurrence of by , “weakly” by “strictly”, and “strong” by “strictly strong”. Q.E.D. i Proof of Theorem 2. Let s := b(c). We first show that s > o for all i. By i i oi i i definition, o ∈ F (c Proof of Theorem 3. Let s := b(c). As above, oN ∈ F (c) implies s = W N F (c) > o . i This time, we show that s 0 oi i N i then s ∈ F (˜ci ) implies s si, which together with o−i 6 o−i 6 s−i implies s i)” by “F (c )”. Q.E.D. Proof of Theorem 4. Because of (1) and (2), almost surely the following 36 0 holds: (4) for all t > 0 and δ > 0, there is some t > t such that for all i ∈ I, both at times t00 = t0 + i and t00 = t0 + N + i, i is the chosen player, i.e., jt00 = i, and `t00 < δ. Assume bt has a positive probability to converge to some a that is not in the weak core. Then we can pick a realization in which (4) holds and t 0 0 0 b → a ≺≺G a for some group G ⊆ I, G 6= ∅, and some a ∈ A with a−G ≡ 0. 1 0 Let δ = 3 min{ui(a ) − ui(a): i ∈ G} > 0. Because of the continuity of u, we can pick some > 0 so that for all a00 ∈ A with d(a00, a) < , 00 |ui(a ) − ui(a)| < δ for all i ∈ G. Because of convergence, we can pick t > 0 t00 00 t00 so that d(b , a) < for all t > t. Then |ui(b ) − ui(a)| < δ and hence 0 t00 00 ui(a ) > ui(b ) + 2δ for all t > t and i ∈ G. Now we pick some t0 > t according to (4). 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