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Efficient Non-Cooperative Provision of Costly Positive Externalities Via 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 2 [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 maxf0; ( 4a−i + 16a−i − 2 − 2a−i)=4g. From here on, we always assume indifference curves are lifted to non-negative values in this way. 4 (a) (b) 1.0 a continuous CCF for 1 1.0 1's indiff. curve through Cournot point a continuous CCF for 2 2's indiff. curve through Cournot point CCF mechanism outcome satisficers' outcome Pareto-efficient line 1's indiff.
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