Journal of Mathematical Economics and Finance

Biannually Volume II Issue 1(2) Summer 2016 ISSN-L: 2458-0813 eISSN: 2458-0813 Journal DOI: http://dx.doi.org/10.14505/jmef Issue’s DOI: http://dx.doi.org/10.14505/jmef.v2.1(2).00

Summer 2016 Volume II, Issue 1(2) Editor in Chief D. Carfì, University of California Riverside, USA University of Messina, Italy

Co-Editors

Contents:

M. Campbell, Aurislink, Israel / USA Chapman University, USA M. Gualdani, George Washington University, USA

Assistant-Editors

Research papers

A. Agnew, California State University Fullerton, USA A. Donato, University of Messina, Italy

Inevitability of Collusion in a Coopetitive Bounded Rational Cournot Model with Increasing Demand

Editorial Coordinators A. Kushner, Russian Academy of Sciences, Russia M. Maroun, University of California Riverside, USA

Editorial Advisory Board

1

T. Arthanari, University of Auckland, New Zealand V. Balan, University of Bucharest, Romania

Michael CAMPBELL Aurislink, Israel / USA Chapman University, USA…7

Separable Preferences over Intertemporal Opportunities

3

Somdeb LAHIRI PD Petroleum University, Gujarat, India …59

B. Blandina, Bocconi University, Italy M. T. Calapso, University of Messina, Italy K. Cvetko Vah, University of Ljubljana, Slovenia K. Drachal, Warsaw Technology University, Poland

A critical analytic survey of an Asymmetric R&D Alliance in Pharmaceutical industry: bi-parametric study case

S. Federico, University of Calgary, Canada G. Fontana, Leeds University Business School, UK G. Giaquinta, University of Catania, Italy S. Haroutunian, Armenian State University, Armenia Z. Ibragimov, California State University Fullerton, USA

2

J. Martinez-Moreno, University of Jaén, Spain R. Michaels, California State University Fullerton, USA F. Musolino, University of Messina, Italy R. Niemeyer, University of California Riverside, USA

4

David CARFÌ University of California Riverside, USA Alessia DONATO University of Messina, Italy

J. Mikes, University of Olomouc, Czech Republic

…21

M. Okura, Doshisha College, Japan K. Oliveri, Tor Vergata University, Rome, Italy D. Panuccio, University of Messina, Italy A. Pintaudi, LUISS Guido Carli University, Italy

Survey papers & Lecture notes

R. Pincak, Slovak Academy of Sciences, Bratislava A. Ricciardello, University of Enna, Italy D. Schilirò, University of Messina, Italy

Differential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms

A. Shelekov, Lomonosov University, Moscow M. Squillante, University of Sannio, Italy A. Trunfio, University of Padua, Italy L. Ungureanu, Spiru Haret University, Romania A. Ventre, University of Naples, Italy L. Verstraelen, Katholieke Universiteit, Belgium

http://www.asers.eu/asers-publishing ISSN-L: 2458-0813 eISSN: 2458-0813 Journal DOI: http://dx.doi.org/10.14505/jmef [email protected] [email protected]

1

David CARFÌ University of California Riverside, USA …85

2

The level of information held by a problem solver influences decision processes Angelarosa LONGO University of Sannio, Benevento, Italy Viviana VENTRE University of Sannio, Benevento, Italy

…71

Call for Papers Summer_Issue 2016 Journal of Mathematical Economics and Finance

Journal of Mathematical Economics and Finance (JMEF) is a biannually peer-reviewed journal of Association for Sustainable Education, Research and Science. Aims and Scope. The primary scope of our Journal is to provide a forum to exchange ideas in economic theory which expresses economic and financial concepts and laws using formal and wellconstructed mathematical reasoning, Mathematical Decision Theory, Game Theory, Functional Analysis, Differential Geometry and so on. Our Journal covers and publishes original researches and new significant results and methods of Mathematical Economics, Finance, Game Theory and applications, mathematical methods of economics, finance and management, Quantitative Decision theory and Risk Theory. The mathematical form of economic and financial laws appears of fundamental importance to the developments and deep understanding of Economics and Finance themselves. Such a translation in mathematical terms can determine whether an economic or financial intuition shows a coherent and logical meaning. Also, a full rational and mathematical development of economic ideas can itself suggest new economic concepts and deeper economic intuitions. Editor invitation. The editors encourage the submission of high quality, insightful, well-written papers that explore current and new issues in Mathematical Economics, Finance, Econophysics, Game Theory and applications, mathematical methods of economics, finance and management, Quantitative Decision theory and Risk Theory and the common grounds between these discipline areas. Submissions to JMEF are welcome. The paper must be an original unpublished work written in English (consistent British or American), not under consideration by other journals. Indexing. JMEF will be indexed in ProQuest, CEEOL, EBSCO and RePEC databases, starting with the first issue released. How to cite. JMEF follows the format of the Chicago Manual of Style, 15th edition, chapter 16 (a brief guide to citation style may be found at http://www.chicagomanualofstyle.org/tools_citationguide.html ) or, in case of LaTeX submission, JMEF cites and refers using BibTeX (a brief guide may be found at http://www.csse.monash.edu.au/documents/bibtex/ ). Refereeing process. Invited manuscripts will be due till November 15th, 2016, and shall go through the usual double blind refereeing process. Deadline for submission of proposals: Expected Publication Date:

15th of November 2016 15th of December 2016

Web: http://www.asers.eu/journals/jmef.html e–mail: [email protected] [email protected] Full author’s guidelines are available from: http://asers.eu/journals/jmef/instructions-for-authors.html

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Journal of Mathematical Economics and Finance

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Journal of Mathematical Economics and Finance

Research papers

Inevitability of Collusion in a Coopetitive Bounded Rational Cournot Model with Increasing Demand Michael CAMPBELL, Aurislink, Israel / USA and Chapman University, USA

…7

A critical analytic survey of an Asymmetric R&D Alliance in Pharmaceutical industry: biparametric study case David CARFÌ, University of California Riverside, USA Alessia DONATO, University of Messina, Italy

…21

Separable Preferences over Intertemporal Opportunities Somdeb LAHIRI, PD Petroleum University, Gujarat, India

…59

The level of information held by a problem solver influences decision processes Angelarosa LONGO, University of Sannio, Benevento, Italy Viviana VENTRE, University of Sannio, Benevento, Italy

5

…71

Journal of Mathematical Economics and Finance

6

Volume II, Issue1(2), Summer 2016

DOI: http://dx.doi.org/10.14505/jmef.v2.1(2).01

Inevitability of Collusion in a Coopetitive Bounded Rational Cournot Model with Increasing Demand Michael Campbell Aurislink, Israel/USA Department of Physics and Computational Mathematics, Chapman University, USA [email protected]

Abstract. A coopetitive model, using the structure formulated by D. Carf`ı, is constructed for a bounded rational Cournot model with increasing demand (as with Veblen goods) and any number of agents. This model has a cooperative strategy parameter that interpolates between perfect competition and collusion. For this model, H. Dixon’s result of the “inevitability of collusion” is demonstrated using a cluster expansion idea from percolation models in statistical mechanics to prove positivity of correlation functions. Specifically, it is shown that every agent’s expected payoff increases as the cooperatively chosen interpolation parameter approaches the value that gives collusion. Therefore agents will cooperatively agree to collude. When the behavior is perfectly rational (zero temperature), collusion does not result in an increase in payoffs since agents produce at maximum output in competition or collusion: agents gain no benefit for putting in the extra effort to collude. So we see that neoclassical analysis (i.e., Nash equilibrium analysis) can not explain collusion in this case. However when we consider the full bounded rational model (positive temperatures), we recover Dixon’s result to see that agents will cooperatively decide to collude to maximize payoffs. We point out that the neoclassical model is the zero-temperature limit of the general bounded rational model utilized here in accordance with the Bohr correspondence principle. Keywords: Cournot Model, Coopetetive Game, Perfect Competition, Collusion, Correlation Inequality, Potential Game, Bounded Rationality, Logit Equilibrium, Nash Equilibrium, Noisy Directional Learning, Statistical Mechanics, Entropy, Cluster Expansion. JEL Classification: C31, C61, C62, C65, C71, C73, C79, D41, D43, D83.

1. Introduction Veblen goods (Veblen 1973), those related to “conspicuous consumption” and status seeking, have the fascinating property that higher prices result in increased demand. This happens because with a higher price, the projected status of the buyer increases. Examples of this are expensive cars and so called California “cult wines”. As a result of volunteering on a large-scale charity wine tasting and auction event, the author (who is also a wine judge) received “inside information” about prestigious wineries selling their lesser quality or excess grapes (which were still very good) to “less prestigious” wineries who make wine that is significantly lower in price. The less presitigious winery agrees to not disclose the prestigious winery as a source of grapes for reputation reasons. In this case, the prestigious winery will

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Journal of Mathematical Economics and Finance

not produce at maximum output because of the desire to maintain high quality standards, yet they may cooperate with other wineries to maximize profit and not waste the lesser quality grapes. Such ideas as protecting the “reputation” of a product are very subjective and often emotionally significant. Consequently, modeling based on rational decisions alone is not adequate to explain such factors. Ideas of “bounded rationality” have been around for a long time (Luce 1959; Simon 1955) and have resulted in explanations of important empirical phenomenon that could not be explained by rational assumptions alone. For example, the “price impact function” (relationship between change in stock price and volume traded) was shown to have the same type of shape for all firms of all sizes provided it is assumed that buying and selling is random (Lillo et al. 2003). In a similar spirit, we’ll show that rational behavior alone is inadequate to explain collusion in a model involving Veblen goods. We wish to consider the effects of such subjective “bounded rational” reasons for not producing maximum output, and the resulting motivation to cooperate to maximize profit. As such, we will look at a bounded rational Cournot model with any number of agents and an increasing linear demand. Such a model was shown have a potential (Monderer and Shapley 1996) and thus to “equilibrate” to a Gibbs measure as mentioned in Campbell 2005, Carf`ı and Campbell 2015. This results from two different approaches: 1. a drift-diffusion model where agents follow the gradient of their payoff function along with a perturbation by a weighted Gaussian white noise process, or 2. (information) entropy of the measure on joint decisions is maximized with the constraint of a constant mean potential. In the first approach (see Rassuol-Agha and Sepp¨al¨ainen 2015 along with Dai and Williams 1995; Dupuis and Williams 1994; Harrison and Williams 1987 and references therein for a complete and rigorous development), the noise represents deviations from rationality (endogenous to each agent) which could be due to experimentation, lack of complete information, preference shocks, behavioral bias, etc. (c.f., Campbell 2005, Carf`ı and Campbell 2015 general approaches that don’t result in the Gibbs measure are Binmore and Samuelson 1997, Binmore et al. 1995). In the second approach (see Rassuol-Agha and Sepp¨al¨ainen 2015 and Ellis 1995 for a complete and rigorous development), agents arbitrage information out of the system to gain knowledge and improve payoff, thus maximizing Shannon information entropy (Shannon 1948). The constraint of a constant potential is a way to enforce bounded rationality: if agents are perfectly rational, the mean of the potential would be its maximum - the Nash equilibrium. This is the approach of “large deviation theory” in statistical mechanics1 , which introduces a “temperature”, with lower temperature representing more rational behavior. The analysis of a Gibbs measure is, in effect, statistical mechanics. But it is important to point out that a statistical mechanics model is not constructed a-priori as in Durlauf 1996; it is a result of a bounded rational modeling of an existing standard model in neoclassical economics. The usefulness of statistical mechanics is evident here (c.f., also Durlauf 2012), in that behavior that is expected (i.e. collusion, c.f., Dixon 2000), and even observed as pointed out in the cult wine example above, is recovered when neoclassical analysis does not predict its existence. Since we consider a demand that is increasing, the resulting potential is aligning (the term “ferromagnetic” is used in statistical mechanics), which means agents are influenced to 1 Rassuol-Agha and Sepp¨ al¨ ainen 2015 contains a model of a “heat bath” of an “ensemble” of many identical non-communicating systems (useful for games with a small number of agents - an ensemble is essentially an artifact of assuming the existence of a probability measure on the state space of decisions), and Ellis 1995 models a single system with many particles/agents (appropriate when considering a limit of infinitely many agents in a single game) - both produce the Gibbs measure.

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Volume II, Issue1(2), Summer 2016

make a similar type of decision as other agents in the model. The Gibbs measure converges to the Nash equilibrium in the zero-temperature limit of perfectly rational behavior (Campbell 2005, Carf`ı and Campbell 2015). This is the Bohr correspondence principle, which states that a newer model should reproduce the results of the older well-established theories in the appropriate limit (Bohr and Rosenfeld 1976). We also point out that because we consider a finite number of agents, no phase transition occurs. A significant result of Dixon (Dixon 2000) showed for a certain bounded rational model, that under general assumptions, “there are powerful long-run forces pushing the firms [...] towards collusion”. That model analyzed a continuum of (two-agent) duopolies for which the dynamics were determined by “aspiration” functions that determined whether or not an agent experiments with other strategies, or keeps playing the same strategy. Here, agents are allowed to cooperate to alter the type of individual payoff they want to maximize, although their actual payoff function is fixed. For example, agents could choose to collude and maximize industry-wide payoffs, or only maximize their own individual payoff function. Agents are both competing and cooperating, and we are taking a “coopetitive” approach (Carf`ı 2015) rather than an aspirational approach. As such, we see again that Dixon’s result is robust. It was also raised in Campbell 2005 that certain combinations of collusion and competition could exist, and in a footnote there it was pointed out that in a special case, collusion resulted in the same model as perfect competition, but at half the temperature. This implies that collusion is “more rational” behavior and, intuitively, this was motivation to study how agents would behave if they had the ability to choose different levels between perfect competition and collusion. What is interesting about goods with the aforementioned increasing demand, is that maximum output is the Nash equilibrium for individual best-response, as well as for collusion. So the Nash equilibrium analysis of this model, i.e., “zero-temperature” perfectly rational behavior, does not shed any light on collusive behavior for such goods. However when we consider non-rational behavior (i.e., positive temperatures) in addition to rational behavior, we will see that agents will have motivation to collude. This is in part because they don’t produce maximum output on average. Bounded rational reasons for not always producing maximum output could be many workers being absent during a vacation season, manufacturing equipment that is temporarily down for repair, high quality standards as with cult wines, etc. In a sense, the lack of perfect rationality motivates agents to look for a way to cooperate together to increase profit. We will use D. Carf`ı’s approach (Carf`ı 2015) of a coopetitive model to give agents the collective ability to tune their behavior between competition and collusion (see also Carf`ı and Perrone 2013; Carf`ı and Schilir` o 2012b,a,c). We will then show that payoffs increase as the agents increase the competition parameter towards collusion by using a cluster expansion method for continuous spin models (Campbell and Chayes 1998, 1999), thus arriving at Dixon’s important result of the inevitability of collusion for this bounded rational Cournot model. As such, this result demonstrates that the assumption of bounded rationality is essential to predict collusion in certain fundamental models.

2. Coopetitive Cournot Model The simplest oligopoly model is due to Augustin Cournot (Cournot 1838). There is one homogeneous good with demand function p(Q), where Q is the total quantity of the good

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produced. Given an oligopoly of N firms, if each firm produces an amount qi of the good, then N X Q= qi . (1) 1

For the sake of later analysis, each agents’s production qi will be scaled between a minimum production q ≥ 0 and a maximum production q¯ > q. We will assume each agent produces a sufficiently large number of the good so that the qi can be regarded as continuum variables 0 ≤ q ≤ qi ≤ q¯, 1 ≤ i ≤ N . For example, qi = 1 can represent the production of a sufficiently large number of goods. It is noted that smaller production would be handled in the discrete case as in section 4 of Campbell 2005. Each agent uses quantity qi as her strategy based on the payoff function fi (~q) = qi p(Q) − Ci (qi ),

(2)

where Ci is the ith agent’s cost function. We will assume an increasing linear (inverse) demand function as found with Veblen goods p(Q) = a +

b Q, N

(3)

with constants a > 0 and b > 0. Notice that b is divided by N so that demand is based on the average production. Thus demand remains finite for large N . To illustrate this, if each firm were to produce q¯, then p(N q¯) = a + b¯ q is well-behaved and non-trivial (i.e., doesn’t go to infinity or zero). Constant marginal costs (i.e., Ci0 = c, for a constant c > 0) will also be assumed, so that Ci (qi ) = cqi + di .

(4)

We define the “external field” variable h := a − c.

(5)

N b X qj + hqi − di . N 1

(6)

The payoff functions are now fi (~q) = qi

In the case of collusion, each agent would try to maximize total industry profits ftot (~q) =

N X i=1

fi =

N N N X X b X qi qj + h qi − di . N i,j=1 1 1

(7)

The Nash equilibrium can be found by noting that since all outputs are non-negative (i.e., qj ≥ q ≥ 0 for j = 1, ..., N ), the derivative of the ith agents payoff N ∂fi b X b = qj + qi + h ∂qi N j=1 N

(8)

will have a maximum rate of increase at maximum output qi = q¯. This is clearly the best response for agent i’s payoff in (6). We will show that the Nash equilibrium is the same for

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Volume II, Issue1(2), Summer 2016

collusion below by setting up a coopetitive model that interpolates between competition and collusion. The assumption will be made that agents entering the market will have increasing payoffs as production increases for small production levels. The author has observed this with cult wineries in California who produce limited supply: production initially goes up to placate overwhelming demand from the “cult” of followers, many of whom are on a multi-year waiting list. Therefore the derivative in (8) should be nonnegative for small values qi ≥ 0, resulting in the condition h := a − c ≥ 0. (9) For the coopetitive model (Carf`ı 2015), we introduce a cooperative strategy parameter that is cooperatively adjusted by all agents in the game 0 ≤ θ ≤ 1,

(10)

where θ = 0 will result in individual competition and θ = 1 yields collusion. We will do this by defining coopetitive payoff functions for each agent, X fi (~q; θ) := fi + θ fj , (11) j6=i

or explicitly, N

fi (~q; θ) = (1 − θ)

b X qi qk + (1 − θ)hqi − (1 − θ)di N 1 N N N X X b X dj . qk − θ qj qk + θh +θ N 1 1

(12)

j,k=1

Agents determine best responses for each θ, and then decide cooperatively on how to choose θ to maximize all of their payoffs. Note that the coopetitive payoff interpolates between competition and collusion: fi (~q; 0) = fi , fi (~q; 1) =

N X

fj .

competition payoff

(13)

collusion payoff

(14)

j=1

The Nash equilibrium qˆiθ for the coopetitive payoff fi (qi ; θ) is at maximum output qˆiθ = q¯,

1≤i≤N

(15)

since all payoff functions have maximum values at maximum output for all agents. The case for θ = 1 is the same, thus showing that the Nash equilibrium for collusion is again maximum output qi = q¯, 1 ≤ i ≤ N . As a result, we see that the Nash equilibrium analysis can not explain collusion for this model - agents would gain nothing from putting in extra effort to cooperate/collude. To see motivation for collusion, we have to look at the full bounded rational model.

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3. Potential Game Model and Bounded Rational Equilibrium We consider a game with a finite number N of agents, and all of these agents belong to the set Λ. At any moment in time, an agent i ∈ Λ can select a strategy variable qi ∈ A. A configuration ~q of the system is any possible state of the system: ~q = (q1 , q2 , . . . , qN ) ,

(16)

where each qi ∈ A. The set of all possible configurations of the game is Y ΦΛ := Ai ,

(17)

i∈Λ

which is called (pure) state space. The Ai := A here is the interval of possible agent outputs, A := [q, q¯]. (18) A potential game (Monderer and Shapley 1996) with potential V (~q) and payoff functions fi (~q), ~q = (q1 , ..., qN ), for each agent i ∈ Λ satisfies, by definition, ∂ q) ∂qi fi (~

=

∂ ∂qi V

(~q).

(19)

The salient point is that, for each i, the gradient of the potential with respect to the variable of agent i is the same as the gradient of the ith agent’s payoff (with respect the ith agent’s variable). Agents would follow the gradient of their payoff function for “myopic decisions” (agents look at the best local choice), and for potentials with an interior maximum, this would lead to the Nash equilibrium (Monderer and Shapley 1996). In the model presented here, each agent has a strategy variable qi . A potential for the coopetitive payoff functions (12) is: V (~q; θ) =

N N N X (1 + θ)b X (1 − θ)b X 2 qi qk + qk + h qk 2N 2N i,k=1

k=1

(20)

k=1

We point out that V (~q; 0) and V (~q; 1) give the competitive and collusion potentials, respectively. The stochastic dynamics are then given by the Itˆo diffusion (Langevin) equations: for 1 ≤ i ≤ N, dqi (t) = ∂q∂ i fi (~q; θ)dt + ν dwi (t) + ri (qi )dyi (t), (21) where wi (t) is a zero-mean, unit-variance normal random variable from a Wiener process (“dwi (t)” is often referred to as a Gaussian white noise), ν is a variance parameter, r(qi ) a one-dimensional reflection vector field, and y(t) a process that only changes on the boundary for reflection. Here we use inward normal reflection on the boundary of the domain. Using the definition of a potential (19), we can rewrite (21) above compactly as ~ dt + νdw(t) d~q = ∇V ~ + r(~q)d~y (t),

(22)

with ~q = (q1 , q2 , . . . , qN ), dw ~ = (dw1 , dw2 , . . . , dwN ), r(~q) a matrix, d~y = (dy1 , dy2 , . . . , dyN ), ~ and ∇V having components consistent with (21). Below, we develop the stochastic dynamical system that yields the Gibbs measure as the stationary measure. See example 3.8 in Kang and Ramadan 2014 and previous arguments for a rigorous proof, and also Dai and Williams 1995; Dupuis and Williams 1994; Harrison and Williams 1987; Kang and Ramadan 2014; Carillo et al. 2001; Mohammed and Scheutzow 2003 for stochastic/reflection and analytic techniques for semigroups and convergence rate bounds.

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Proposition 1 Let ρ (~q) be the joint density function over pure state space ΦΛ for a potential game with a finite number of agents N and potential V . Consider the dynamics 2 ~ dt + νdw(t) d~q = ∇V ~ + ~r(~q)d~y (t),

(23)

~ = (∂V /∂q1 , ∂V /∂q2 , . . . , ∂V /∂qN ), w where ~q ∈ ΦΛ , ∇V ~ a vector of 2N standard Wiener processes which are identical and independent across agents and time. Furthermore, the wi have mean zero and variance one with reflecting 3 boundary conditions regulated by the (normal) direction field ~r and the process ~y (t). If the process ~q(t) satisfies the dynamics of (21) (i.e., is a solution to the associated submartingale problem of the reflected diffusion), then the stationary density satisfies the Fokker-Planck equation ∂ρ(~q, t) ~ · [∇V ~ (~q; θ) ρ(~q, t)] + = 0 = −∇ ∂t

ν2 2 q , t) 2 ∇ ρ(~

(24)

and hence the stationary measure for variance ν 2 is the Gibbs state (Kang and Ramadan 2014 corollary 1 and example 3.8)
exp ν22 V (~q; θ)
ρθ (~q, t) = ρθ (~q) = R . (25) exp ν22 V (~q0 ; θ) d~q0 ΦΛ In statistical mechanics, the term in the exponent of (25) is −E(~q)/(kT ), where k is Boltzmann’s constant, T is temperature, and E(~q) is the energy of configuration ~q. Hence the analogy of a potential game to statistical mechanics is that ν 2 (deviation from rationality; influence of the noise in dynamics (21)) is proportional to ‘temperature’ and the potential V is the negative ‘energy’ of the system (Campbell 2005). We point out that maximums of the potential (20) are the appropriate refinement of the Nash equilibriums for a potential game (c.f., Carbonell-Nicolau and McLean 2014). An “observable” g ∈ C(ΦΛ ) is a continuous function on pure state space such as a payoff fi (~q), inverse demand p(~q), quantity produced by a particular agent qk , etc. We denote the expected value of an observable g with respect to the Gibbs state (25) by Z hgiθ := g(~q)ρθ (~q)d~q. (26) ΦΛ

4. Expected Payoffs, Coopetition, and Correlation Functions We follow the procedure for the coopetitive model (Carf`ı 2015) to determine behavior of agents. First, agents determine the equilibrium for a given cooperative parameter θ (in Carf`ı 2015 it is the Nash equilibrium, which is the Gibbs equilibrium at zero temperature). As we saw in the last section, that equilibrium for all agents in the bounded rational model will be the Gibbs state ρθ (~q). Now agents will cooperatively decide which value of θ to play, based on the best outcome. That is, agents will see how their payoffs change with respect to the cooperative parameter and then cooperatively decide on a specific value for θ that will be the most beneficial to all agents. As such, we need to determine how expected payoff changes 2 3

Note that agents are only playing pure strategies in these dynamics. We use a normal reflection vector field on the boundary ∂ΦΛ as in Kang and Ramadan 2014.

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Journal of Mathematical Economics and Finance

with respect to θ. First, we define the partition function as Z Zθ =

exp [βV (~q; θ)] ΦΛ

N Y

dqi

(27)

i=1

Z =

exp (βV (~q; θ)) d~q,

(28)

ΦΛ

where inverse temperature β is defined as the inverse of temperature T and from (25), 1/T = β := 2/ν 2 , and a convenient notation

N Y

d~q :=

(29)

dqk

(30)

k=1

is used. The change in payoff with respect to the cooperative parameter θ is R d ΦΛ exp (βV (~q; θ)) fi (~q)d~q d hfi iθ = dθ dθ Zθ

∂V  

− hf i = fi ∂V i θ ∂θ θ , ∂θ θ

(31) (32)

which is a correlation function. We will see that this type of correlation will be positive if the “energy-energy” correlations are positive (this is the term used in statistical mechanics; cf. Campbell and Chayes 1998 for such correlations in “continuous spin” models). It is helpful to change variables in the partition function so that the strategy interval A = [q, q¯] in (18) is centered at the origin. To this end, let γ := (¯ q + q)/2,

(33)

κ := q¯ − q.

(34)

The translated strategy variable is now ui := qi − γ,

(35)

and this translated strategy variable is in the interval ui ∈ A˜ := [−κ/2, κ/2].

(36)

The potential (20), with the translated strategy variables, has the form V (~u; θ) =

N X

(1+θ)b 2N

ui uk +

(1−θ)b 2N

i,k=1

N X

h u2k + (1 + θ)bγ +

(1−θ)bγ N

k=1

+

N iX +h uk

(37)

k=1

(1+θ)b 2 γ N 2

+

(1−θ)b 2 γ 2

+ hγN

The partition function can now be written in translated form (c.f. Appendix B Campbell 2005) as Zθ = exp

h

β(1+θ)bγ 2 N 2

+

β(1−θ)bγ 2 2

 × exp β (1+θ)b 2N

N X j,k=1

+ βhγN

uj uk + β

iZ

N Y

duj

[−κ/2,κ/2]N j=1 N X (1−θ)b 2N j=1

 u2j + β (1 − θ)bγ +

(1−θ)bγ N

 N X +h uj  . j=1

(38)

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Volume II, Issue1(2), Summer 2016

The payoff from (6) and derivative of the potential in (37) have translated forms fi (~u) =

N N b X bγ X ui uk + (bγ + h)ui + (bγ 2 + hγ − di ) uk + N N

and b ∂V (~u; θ) = ∂θ 2N

(39)

k=1

k=1

N X

ui uk +

i,k=1;i6=k

N (N − 1)bγ X (N − 1)bγ 2 uk + . N 2

(40)

k=1

To accomodate the analysis of correlations, we “duplicate” the variables ui in (36) with vi ∈ A˜ = [−κ/2, κ/2].

(41)

The correlation can then be written



∂V  fi ∂θ θ − hfi iθ ∂V Z ∂θ Z θ  1 eβV (~u;θ)+βV (~v;θ) [fi (~u) − fi (~v )] ∂V u) − = 2 ∂θ (~ 2Zθ ˜2N A

∂V ∂θ

 (~v ) d~u d~v

(42)

For correlations, it will be easier to rotate variables from (ui , vi ) ∈ A˜2 to (xi , yi ) ∈ Di , where for the agents i (1 ≤ i ≤ N ) in the set Λ, yi = (ui − vi ) /2,

xi = (ui + vi ) /2,

(43)

and the rotated duplicated variables are in the diamond set Di = D := {k(xi , yi )k1 = |xi | + |yi | ≤ κ/2} .

(44)

The full rotated, duplicated configuration space is then Y ΥΛ := Di = DN ,

(45)

i∈Λ

and the combined potentials V (~u; θ) + V (~v ; θ) in (42) become V (~x; θ) + V (~y ; θ) =

N X

(1+θ)b N

(xj xk + yj yk ) +

(1−θ)b N

j,k=1

N X

(x2k + yk2 ) + 2h

k=1

N X

xk ,

(46)

k=1

where we ignore constant terms since they are cancelled out in (42) by the square of the partition function in the denominator (c.f. equation (38)). The difference functions in (42) have rotated forms fi (~x) − fi (~y ) =

2b N xi

N X

yj +

j=1

(~x; θ) −

∂V ∂θ

(~y ; θ) =

N X

xj +

2bγ N

j=1

and ∂V ∂θ

2b N yi

b N

N X i,k=1;i6=k

N X

yj + (bγ + h)yi ,

xi yk +

(N −1)2bγ N

N X

yk .

k=1

We point out that all terms in (46), (47), and (48) have nonnegative coefficients.

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(47)

j=1

(48)

Journal of Mathematical Economics and Finance

The exponential in (42) can be expanded in a Taylor series

∂V 

 fi ∂θ θ − hfi iθ ∂V ∂θ θ ZZ ∞ X  βm m = [V (~x; θ) + V (~y ; θ)] [fi (~x) − fi (~y )] ∂V x) − 2 ∂θ (~ 2m!Z DN θ m=0

∂V ∂θ

 (49) (~y ) d~x d~y ,

and since β ≥ 0, it only needs to be shown that the integrals in (49) are positive. To that end, an idea from graphical representations of continuous spin models will be used (Campbell and Chayes 1998, 1999). We represent each continuous “spin” xi and yi as an Ising ±1 spin multiplied by an interaction, xi = σi si ,

σi = ±1,

si = |xi |,

(50)

yi = ωi wi ,

ωi = ±1,

wi = |yi |,

(51)

where the interactions are integrated over the positive quadrant of the diamond (44) D+ := {(si , wi ) | si ≥ 0, wi ≥ 0, k(si , wi )k1 ≤ κ/2} .

(52)

Configurations of the graphical representation variables are denoted ~σ := (σ1 , σ2 , ..., σN ) ∈ {−1, +1}N , ~s := (s1 , s2 , ..., sN )

(53) (54)

N

ω ~ := (ω1 , ω2 , ..., ωN ) ∈ {−1, +1} , w ~ := (w1 , w2 , ..., wN ), where (~s, w) ~ :=

N Y

(si , wi ) ∈ D+


N

(55) (56)

.

(57)

i=1

The integrals in (49) have the following form in the graphical representation variables: ZZ X X Y Y Y m i Y ni d~s dw ~ σ i1 ωi2 si3 3 wi4 4 . (58) + N (D )

~ σ ∈{−1,1}N ω ~ ∈{−1,1}N i1

i2

i3

i4

If a single spin σi or ωi appears in the product (58), summing over σi = ±1, ωi = ±1 will result in a value of zero. If the spin products are empty (i.e., equal to one), then the result is an integral of the variables si and wi to various powers. Since the si and wi on nonnegative N on their domain (D+ ) , we see that the integral in (58) is positive, and thus the correlation in (49) is positive. All of this was a notationally heavy way to show positivity by symmetry, but the techniques above are useful in more general situations (Campbell and Chayes 1998, Campbell and Chayes 1999). As a result, from (32) it is seen that the derivative of any agent’s expected payoff function hfi iθ is strictly increasing in the cooperative parameter θ. Agents will therefore decide cooperatively to choose the maximum value of θ = 1, which we see is collusion from (12) and (14)

Conclusion Upon analyzing a Cournot model with increasing demand as found in Veblen-type goods, we saw that standard neoclassical theory using the Nash equilibrium was not sufficient

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to explain any collusion, since output would always be at the maximum for any degree of collusion - there would be nothing to gain if agents put in extra effort to collude. The fact that collusion does happen with such goods, as with the California cult wines, was motivation to approach such a market with a bounded rational approach. It was previously observed that collusion is in a sense, more “rational” behavior than individual competition, in the sense that the degree of non-rational behavior is effectively lowered as a result of colluding. That observation was formalized here in a more sophisticated sense, by giving agents the ability to cooperatively decide on various degrees of collusion through the use of a coopetitive model. Accordingly, agents will decide to collude to the fullest extent possible, which results in maximum expected payoffs for every agent. The approach here is different from the aspirational model of Dixon which examined many markets with two agents - we look at a single market with any number of agents. But the end result is the same. This shows a robustness of Dixon’s result on the “inevitability of collusion”, as well as the utility of coopetitive models. A useful generalization would be to analyze decreasing demand Cournot models which have anti-aligning, negative interactions. The approach used here does not work for such “antiferromagnetic” models, so another approach to proving positivity of correlation functions would be needed in that case.

Acknowledgments. The author wishes to thank Hugh Dixon, Ruth Williams, Jim Dai, J. Michael Harrison, and David Carf`ı for useful comments and suggestions.

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Carf`ı, D. and M. Campbell (2015). Bounded Rational Speculative and Hedging Interaction Model in Oil and U.S. Dollar Markets. Journal of Mathematical Economics and Finance 1 (1), 4–28. http://dx.doi.org/10.14505/jmef.v1.1(1).01. Carf`ı, D. and E. Perrone (2013). Asymmetric Cournot Duopoly: A Game Complete Analysis. Journal of Reviews on Global Economics 2, 194–202. https://dx.doi.org/10.6000/ 1929-7092.2013.02.16. Carf`ı, D. and D. Schilir` o (2012a). A Model of Coopetitive Game for the Environmental Sustainability of a Global Green Economy. Journal of Environmental Management and Tourism 3 (1(5)), 5–17. Carf`ı, D. and D. Schilir` o (2012b). A coopetitive model for the green economy. Economic Modelling 29 (4), 1215–1219. https://dx.doi.org/10.1016/j.econmod.2012.04.005. Carf`ı, D. and D. Schilir` o (2012c). Global Green Economy and Environmental Sustainability: A Coopetitive Model. In S. Greco, B. Bouchon-Meunier, G. Coletti, M. Fedrizzi, B. Matarazzo, and R. Yager (Eds.), Advances in Computational Intelligence (14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, Catania, Italy, July 9-13, 2012, Proceedings, Part IV), Volume 300 of Communications in Computer and Information Science, pp. 593–606. Springer Berlin Heidelberg. https://dx.doi.org/10.1007/978-3-642-31724-8_63. Carillo, J. A., A. J ungel, P. Markovich, G. Toscani, and A. Unterreiter (2001). Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatshefte fr Mathematik 133 (1), 1–82. Cournot, A. (1838). Researches Into the Mathematical principles of the Theory of Wealth. London: Macmillan & Co. Dai, J. G. and R. J. Williams (1995). Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. Theory of Probability and Its Applications 40, 1–40. Correctional note, 50, 346–347, (2006). Dixon, H. (2000). Keeping up with the Joneses: competition and the evolution of collusion. Jour. of Econ. Behavior & Organization 43, 223–238. Dupuis, P. and R. J. Williams (1994). Lyapunov functions for semimartingale reflecting Brownian motions. Annals of Probability 22, 680–702. Durlauf, S. (1996). Statistical Mechanics Approaches to Socioeconomic Behavior. SSRI reprint #455 . Durlauf, S. (2012). Complexity, economics, and public policy. Politics, Philosophy & Economics 11 (1), 45–75. Ellis, R. S. (1995). An overview of the theory of large deviations and applications to statistical mechanics. Scandinavian Actuarial Journal 1995 (1), 97–142. Harrison, J. M. and R. J. Williams (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Annals of Probability 15, 115–137. Kang, W. and K. Ramadan (2014). Characterization of Stationary Distributions of Reflected Diffusions. Annals of Applied Probability 24 (4), 1329–1374.

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