On the stability of Nash Equilibria in super(sub)-modular games : an evolutionary perspective

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On the Stability of Nash Equilibria in Super(Sub)-Modular

Games: an Evolutionary Perspective

Adrienne Goedhart∗† October 23, 2014

Abstract

The aim of this paper is to extend the evolutionary-model of Hommes, Ochea and Tu-instra (2011) to a broader class of games, the so-called supermodular and submodular games. The dynamics of the interaction between the different heuristics are investigated for a fixed fraction and a evolving fraction. The development of the fraction depends on how profitable that certain heuristic is. The heuristics that are considered are the Naive and Nash heuristic. Overall, it holds that supermodular games have more stable dynamics than submodular games.

It is not straightforward that games with complements or substitutes lead to a supermod-ular or submodsupermod-ular game. In general it holds that the combination of demand function, cost function and the parameter values determines whether a game is supermodular or subdmodular.

Keywords: Supermodular, Submodular, Evolutionary competition, Stability of Nash equilibrium

Thesis supervisor: dr.M.I. Ochea

∗

10658343

†

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1 Introduction

Theocharis (1960) examined the stability of a steady state in a n-firm oligopoly model with a linear demand function and constant marginal cost. He uses a discrete-time Cournot (1897) adjustment process which implies that every firm assumes that the other firms produce the same quantity as they did in the previous period. He displayed that the stability of the steady state relies on the number of firms in the model. For a monopoly and duopoly there exist one stable steady state but if there is a trio-poly, bounded but perpetual oscillations are obtained. For more than four players, the steady state loses stability and quantities diverge.

Fisher (1961) showed that the dynamics of the model with discrete-time Cournot ad-justment process can be stabilized when the marginal cost function are increasing and the adjustment coefficient is small. Moving from a discrete-time adjustment process to a continuous-time adjustment process both McManus & Quandt (1961) and Hahn (1962) showed that the equilibrium is stable under general conditions.

Hommes, Ochea and Tuinstra (2011) generalize the model of Theocharis (1960) to a evolutionary version. In the evolutionary version they do not only consider the Cournot heuristic (best response to naive expectation) but also the rational and Nash heuristic (expectation rules concerning the aggregate output of rivals). The faction of firms who uses a certain heuristic is variable over time and depends on the performance of that specific heuristic. Although perpetual fluctuations are still obtained, Hommes, Ochea and Tuinstra showed that the introduction of multiple heuristics tends to stabilize dy-namics.

In this paper the evolutionary investigation of the stability of an equilibrium in a n-player game is extended to a larger class of games, namely games of strategic complements and substitutes (Bulow et al., 1985). We examine the interaction between the naive players, Nash players and imitators. Just like Hommes, Ochea and Tuinstra (2011) we no longer consider homogeneous expectations but heterogeneous expectations.

Strategic complements roughly means, the best response of any firm is increasing in the action of other firms. Supermodular games (Vives, 1990) are a suitable strategic model of interaction between agents in presence of strategic complements.

The theory behind supermodular games is based on lattice theory and monotonicity properties. A major advantage of supermodular games is that it is not necessary to show that the profit function is quasi-concave, in order to prove that there exist a fixed point which maximizes profit. This was verified by Topkis (1979) who showed that ev-ery supermodular game has an nonempty set of solutions which always consist out of a largest and a smallest solution. The non-emptiness of the set of solutions was proved with the aid of Tarski’s fixed point theorem (1955). This theorem states that when a function is non-decreasing, there exist a set of fixed points which consist out of a largest and smallest element. Since supermodularity practically yield monotone non-decreasing reaction curves in the actions of other players, Tarski’s theorem can be applied. Another advantage of supermodular games is that they have nice stability properties in contrast to games with non-monotone best responses. The dynamics of games with non-monotone

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best responses can become quite chaotic. An almost similar concept to supermodularity is increasing differences. Supermodularity is the strongest concept of the two, however if the functions are defined on a product of ordered sets, which is frequently the case for strategies chosen by the agents, the two concepts correspond (Topkis, 1978). In general it holds that supermodular games are more handy to work with, while increasing differ-ences are more recognizable.

A closely related subject to games with strategic complementarities are games with strategic substitutes (Vives, 1990). It is not literally stated that games with strate-gic substitutes are captured by submodular games, but the two concepts are certainly related to each other. Submodular two player games can be made supermodular by reversing the strategies.

A game with strategic substitutes involves a game where an increase in one player’s strat-egy makes the best response of other players decrease. This kind of interaction arises in most cases under imperfect competition. Examples of imperfect competition are price leadership and games with negative externalities at the margin (e.g. Congestion games, the payoff of these games depends on resources which are limited and the number of players who uses these resources). The methods behind strategic substitutes are also based on lattice theory. Only little is known about these games and the existence of a equilibrium is not as straightforward as for supermodular games (Jensen, 2005). Vives states the following about a Pure Strategy Nash Equilibrium (PSNE) in submodular games:”When best replies are decreasing existence is guaranteed for two player games but not in general. In the particular case of one-dimensional strategy sets where the best reply of a player depends only on the aggregate actions of others existence can be shown for any number of players”(Vives (2001), p.43). Here the aggregate actions of others means the linear sum of the actions. This existence theory is extended by Dubey et al. (2006). They allow for best replies which depend on a function of the other actions, which is not necessarily a linear sum like Vives stated.

Exploring the supermodular games in a dynamic environment is an area where so far not much research has been done. However only Schipper (2002) explored the interaction between different heuristic for aggregative quasi-submodular games and proved that imi-tators (these agents mimic the action of the most succesful player in the previous round) are strictly better of than best response players in a Cournot oligopoly. He also provides an evolutionary foundation for aggregate-taking or Walrasian behavior for a submodular game with imitation rules (Schipper, 2004).

In this paper the dynamics of the Nash, Naive and Imitation heuristic are investigated for supermodular and submodular games. The fraction of players who follow a certain heuristic can be fixed or evolves over time. The development of the fraction depends on how profitable that certain heuristic is.

Section 2 gives an mathematical introduction to supermodular and submodular games, Section 3 shows some practical examples which lead to supermodularity or submodular-ity, Section 4 gives a short introduction to all that needs to be knows about evolutionary dynamics, Section 5 illustrates the local dynamics for the supermodular games while

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Section 6 shows the dynamics for the submodular game. Finally, a conclusion is drawn.

2 An introduction to Supermodular and Submodular games

Supermodular games can be defined as games with as main feature: the best response of any firm is increasing in the actions of the other firms. Decisions of two or more players mutually reinforce one another. This phenomenon is called strategic complements. On the contrary games with stategic substitutes are games where an increase in the strategy of one player will lead to a decrease in the strategy of another player.

A supermodular (and submodular) game can also be expressed in terms of the cross partial derivative of the profit function.

Since most examples of supermodular and submodular games fall within the field of industrial organization, we give a small introduction about Cournot and Betrand com-petition with as market structure an oligopoly. Note that supermodular and submodular games could also capture fields like politics and search theory.

Consider a model with n firms, between the firms there can be Cournot competition or Bertrand competition. For a Cournot competition the companies decide how much output (Qi(p)) they produce and the corresponding price is set such that the demand is

equal to the total quantity. In a Bertrand competition the companies set prices (Pi(q))

and the corresponding quantity will be determined by the demand of the customers. For Cournot the immediate profits looks as follows: Πi = P (qi, Qi)qi− Ci(qi), where Qi

denotes the quantities set by all firms except firm i and P is the inverse demand function. In the case of constant marginal costs the profit function becomes Πi = (pi− ci)qi. The

Bertrand model has the following profit function: Πi = piD(pi, Pi) − Ci(qi), where Pi

denotes the prices set by all firms except firm i and D is the demand function. In the case of unit costs the profit function is Πi= (pi− ci)qi.

Since we want to maximize profits we obtain the following first order conditions: Cournot : ∂Πi ∂qi = P (qi, Qi) + P0(qi, Qi)qi− Ci0(qi) = 0 (1) Bertrand : ∂Πi ∂pi = D(pi, Pi) + piD0(pi, Pi) = 0 (2)

The first order conditions give rise to the reaction curve/best-response correspondence, which for Bertrand is pi = R(Pi) and for Cournot it is qi= R(Qi) .

The first order condition does not always gives an explicit solution of the reaction curve, sometimes the maximization problem implicitly defines the reaction curve. For now this is not an issue since the main interest is the supermodularity/submodularity property which involves the slope of the reaction curve and not the reaction curve itself, which will be explained in more detail subsequently.

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Definition Supermodular Game

Before the definition of a supermodular game is stated, first some assumptions need to be made about the game. Consider a profit function πi which is twice continuously

differentiable and assume that Ai, where Ai denotes the strategy set of player i, (e.g.

for a Cournot game Ai contains all quantities the firm can possibly set) is the cartesian

product of compact intervals of the reals. Then a game is supermodular if (Vives, 1990): • ∂2πi

∂ai,h∂ai,k ≥ 0∀k 6= h. Strategic complementary in own strategies ai

• ∂2πi

∂ai,h∂aj,k ≥ 0∀j 6= i and for all h and k. Strategic complementary in strategies of

the rivals ai1

The first item implies that the marginal profit of action h of player i is increasing in the other strategies of player i. The second item means that the marginal profit of action h of player i is increasing in the actions of the opponents.

These two properties have influence on the slope of the reaction curve. Take for example a Cournot Game, from the first order condition (1) it follows that:

∂r(Q i) ∂Q i = − P0(qi, Qi) + qiP00(qi, Qi) 2P0_(q i, Qi) + qiP00(qi, Qi) − C00(qi) = −∂ 2_Π i/∂qi∂Qi ∂2_Π i/(∂qi)2

This example can be generalized. Overall, the slope of the reaction curve can be written as: ∂r(a i) ∂aj = −( ∂ 2_π i ∂ai∂aj )/ ∂ 2_π i (∂ai)2 (3) The denominator ∂2πi

(∂ai)2 is negative if we have a maximization problem. If the game is

supermodular we have ∂2πi

∂ai∂aj ≥ 0 which in turn ensures that the reaction curve of the

firms is increasing. In general it holds, if ∂2πi

(∂ai)2 < 0, the sign

∂2_π

i

∂ai∂aj= sign ∂r(a i)

∂aj .

Note the following two items are not the same: • ∂2πi

(∂ai)2 < 0

• ∂2πi ∂ai,h∂aj,k < 0

The first one means that the marginal profitability in a firm is decreasing in the prices charged for the same product by the same firm. The second item shows the case when we are dealing with a multi-product firm. This item means that the marginal profitability

1_a

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is decreasing in the prices for other products charged by the same firm.

Supermodular games have some nice properties. Every supermodular game has an nonempty set of solutions which consist out of a smallest and largest element. The non emptiness of the set of solutions was proved with the aid of Tarski’s fixed point the-orem (1955): ’If T is a complete lattice and f :T→ T is a nondecreasing function, then f has a fixed point. Morever, the set of fixed points of f has sup{x ∈ T |f (x) ≥ x}as its largest element and inf{x ∈ T |f (x) ≤ x}as its smallest element ’. Since supermodularity practically yields monotone non-decreasing reaction curves in the actions of other play-ers, Tarski’s theorem can be applied. The monotone increasing reaction curves can be guaranteed in supermodular games since the marginal benefit of increasing the strategy of a player raises with the levels of the strategies of the rivals.

For a 2-player game it does not matter if the reaction curve has an increasing or decreas-ing selection since the composite best reply map, which is necessary to find a intersec-tion of the two reacintersec-tion curves, will be an increasing funcintersec-tion (Vives, 1990). Therefore Tarski’s theorem can always be applied when there are 2 players involved.

Besides the guarantee of an equilibrium in a supermodular games, the equilibrium points also have nice stability properties. Before we state the theorem which contains the nice stability properties, we first define what a tˆatonnement is. A discrete tˆatonnement is an adjustment process which reaches an equilibrium and examine the stability of that equilibrium. E.g. consider a duopoly, in this adjustment process each firm takes turn in adjusting it’s output optimally given the output of the other firm.

The following theorem about supermodular games and stability was stated by Vives (2001):

Let G be a supermodular game with continuous payoffs:

(i) Then any tˆatonnement at approaches the set [a, a] where a and a are respectively, the smallest and largest equilibrium points of the game.

(ii) Then, if the players always select the largest (or the smallest) best response (or, alternatively, if G is strictly supermodular- and we make the convention that if the rivals of player i choose the same strategies in t and t+1, then player i also chooses the same strategy in t+2 as in t+1) then a tˆatonnement starting at any initial value a0 in A+(A−) converges monotonically downward (upward) to an equilibrium point in the game. Corollary, if the equilibrium of a supermodular game with continuous payoffs in unique, it is globally stable.

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Figure 1: Tˆatonnement in a supermodular game, with best reply functions r1(.), r2(.)

The theorem states that in a supermodular game almost all adaptive learning processes converge to a subset of undominated strategies (Milgrom and Roberts, 1990), which is bounded from above and below by the largest and smallest equilibrium point, respec-tively. Within the undominated strategies subset, the product best-reply correspondence of the players, independent of a0∈ A− or a0 ∈ A+_{, will convergence towards an}

equilib-rium point.

Definition Submodular Game

Unfortunately it is not certain from the literature if a submodular game can be defined in terms of the cross partial derivative of the profit function in the literature. The concept is only explained in view of lattice and order theory. It is not straightforward whether games with strategic substitutes are captured by submodular game but since the two are so closely related, one could reason as follows:

Submodular games gives a best response correspondence which is decreasing in the strategies of the opponents. It is shown that the response curve can be explained in terms of the derivatives of the profit function. If the goal of the game is profit max-imization, the denominator is always negative. This results leads to the fact that the sign ∂2πi

∂ai∂aj= sign ∂r(a i)

∂aj . From here it could be suggested that

∂2_π

i

∂ai∂aj is negative since the

reaction curve is decreasing for a submodular game. It is a fact that strategic substitutes lead to ∂2πi

∂ai∂aj < 0.

So from here we could we assume that submodularity and games with strategic substi-tutes both lead to:

• ∂2πi

∂ai∂aj ≤ 0

All examples of submodular games which are introduced in the next section fall within a certain subgroup of submodular games, namely aggregative quasi-submodular games. Aggregative games (Corchon, 1994) are games where the payoff function of each player

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depends on the aggregator (function of all strategies) and it’s own strategy. For this specific subgroup we do know that the cross-partial derivative of the profit function is negative (Topkis ,1998) and therefore the reaction curve is downward sloping.

3 Examples of Supermodular and Submodular game

In all upcoming examples it is assumed that we have a one dimensional strategy set. This means that there is no multiproduct firm. Therefore the focus lies on the cross partial derivative: ∂2πi

∂ai∂aj and not on the other derivative:

∂2_Π

i

∂ai,h∂ai,k.

The analysis starts by interpreting the cross partial derivative of the profit function. In some cases the profit function is given in others it is derived by considering different combinations of demand and cost functions, and these combinations will lead to an ex-ample of a submodular of supermodular game.

Before all examples are introduced, we want to emphasize that one can not state that a game with goods which are substitutes or complements lead to a supermodular or sub-modular game, respectively. It is the combination of the demand and cost function plus the parameter values which determines the supermodularity or submodularity property.

3.1 Examples of Supermodular 2-player game

Milgram and Roberts (1990) stated a couple of examples of supermodular games in their paper. The main features of those examples are

• each firm has a unit cost function • the goods are substitutes

• the second derivative of the demand is a non- increasing function in the other firms prices

The last requirement can be written as ∂2_{log D/∂p}

n∂pm ≥ 0 (Note that in this final

condition it is an option to eliminate the logarithm function since it is an increasing transformation, the log function is used here for convenience).

Examples of demand functions that satisfy the final condition according to Milgram and Roberts (1990) are:

• Logit function: Dn= kn/P_j∈Nkjexp[λ(pj − pn)] where λ < 0 and kj > 0 ∀j

• Transcendental Logarithmic function: log(D_n) = αn+P

j∈Nβnjlog(pj)

+P

j∈N

P

i∈Nγijnlog(pi)log(pj), where βnn< 0, γnnn < 0 ∀ n, βjn≥ 0 ∀j and γijn ≥ 0

∀ j 6= i.

• Constant Elasticity of Substitution (CES) function: Dn= y ∗pr−1n /

P

j∈Nprj, where

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The Logit, Transcendental Logarithmic and CES demand function in combination with a unit cost function will lead to a supermodular game. These are all examples of Bertrand competition.

Another example of Bertrand competition which lead to a supermodular game is a Bertrand game with a linear demand function and a convex cost function (Vives, 2001). Consider a linear demand function Qi(P ) = Di(P ) = b − a ∗ pi +Pi6=jcj ∗ pj where

b, a, cj > 0 ∀ j (here P is the vector containing all prices). The profit function of firm i

is given by πi = Qi(P )pi− C(Qi(P )), where we assume that C(Qi(P )) is a convex cost

function . This game will give rise to a game with substitutes.

All examples above fall within the field of industrial organization but there are examples of submodular games which examine other structures. An example of such a structure is Search theory which studies models with buyers and sellers who can not immediately find a business partner. An example of a such a game is a Diamond-type Search Model (Milgrom and Roberts, 1990). In this model there are a finite number of n players who practice effort searching for trading partners. The probability of a trader to find another trading partner depends on the proportion between his own effort and the total effort of the others. Let x(n) ∈ [0, ¯x] denote the effort of player n. Then, the payoff to player n is defined by:

fn(x) = αx(n)

X

m6=n

x(m) − C(x(n))

Here α > 0 and C(.) is the cost function.

3.2 Results Cross-partial derivative for Supermodular games

For the first three examples, for simplicity the analysis is performed by starting with a model which consist out of two firms (n = 2). In some cases the equations are too complicated to solve analytically, here Matlab is used to study the second derivative. For the supermodular examples it is expected that the cross partial derivative is positive. The first supermodular example with the Logit demand function, has the following second derivative: ∂2log Π1 ∂p1∂p2 = k1k2λ 2_exp[λp 1] exp[λp2] (k1exp[λp1] + k2exp[λp2)2 2

Since λ2, k1, k2 are positive and the exponential power always has an positive outcome

it follows that the slope of the reaction curve is positive, which is consistent with what was expected of a supermodular game.

2

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For the Transcendental Logarithmic function, the following second derivative is found: ∂2log Π1 ∂p1∂p2 = γ12 1 p1 1 p2 3

From the derivative we can immediately conclude that the derivative is always positive since γ12 is positive. Overall, the slope decreases when the price increases and the slope

is higher for small prices.

The CES demand function with a unit cost function has the following second derivative: ∂2_{log Π} 1 ∂p1∂p2 = r2p r−1 1 pr−12 pr₁+ pr₂ 4

As expected the second derivative is always positive since r2 is positive and prices are non-negative. Milgram and Roberts (1990) state that r should be smaller than zero for the supermodularity feature, this has to do with the first property of supermodu-lar games: ”Strategic complementary in own strategies ai”. For the second property

”Strategic complementary in strategies of the rivals ai” the value of r does not matter

since the cross partial derivative is always positive independent of the value chosen for r.

The cross-partial derivative for the last Bertrand game with gross substitutes for n firms is given by:

∂2_π i ∂pi∂pj == ∂Qi(P ) ∂pj (1 − ∂ 2_C(Q i(P )) ∂pi∂pj ∂Qi(P ) ∂pi ) +∂ 2_Q i(P ) ∂pi∂pj (pi− ∂C(Qi(P )) ∂pi )5 The first term ∂Qi(P )

∂pj is positive since we have products that are substitutes. The

sec-ond term (1 −∂2C(Qi(P ))

∂pi∂pj

∂Qi(P )

∂pi ) is positive since the cost function is convex (the second

derivative of a convex cost function is positive), and ∂Qi(P )

∂pi is negative.

The last term is equal to zero since ∂2Qi(P )

∂pi∂pj is zero. So it holds that

∂2_π

i

∂pi∂pj is positive

which is in accordance with the properties of a supermodular game.

The cross-partial derivative of the last supermodular example, the Diamand-type Search model, is given by:

∂2fn(x) ∂x(n)∂x(m) = ∂2(αx(n)P m6=nx(m) − C(x(n))) ∂x(n)∂x(m) = ∂(α P m6=nx(m) − C0(x(n))) ∂x(m) = α

3_{See section 8.1, p.47 (Appendix) for the derivation} 4

See section 8.1, p.48 (Appendix) for the derivation

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Since α > 0 the cross-partial derivative is posititive and this is indeed an example of a supermodular game.

3.3 Examples of Submodular 2-player games

According to Schipper(2004) the following examples give rise to a submodular game: • Cournot oligopoly with differentiated substitute products. The price function that

firm i is facing is: pi(qi, t) = βθ(P_j∈Nq_jβ)θ−1q_iβ−1 with 0 < βθ < 1, θ < 1 and

1 ≥ β ≥ 0. (The aggregator is t=P

j∈Np

β

j)

• Betrand oligopoly with differentiated complementary products. Where the demand function looks as follows: Di(pi, t) = (βθ)1/(1−βθ)

p1/(β−1)_i (P j∈Np β/(β−1) j )k , with β < 0, 0 < βθ < 1, θ < 1 and k = _1−βθ1−θ . (The aggregator is t= P

j∈Np

β/(β−1)

j .)

• Common-Pool Resource Game

In the Common-Pool Resource Game each player has an investment e ∈ R++which

he or she can invest in an outside activity with a constant marginal payoff c ∈ R++

or in the common-pool resource. If si ∈ [0, e] is what every person invest in the

common-pool, the return of this investments is: si n P j=1 sj [α n P j=1 sj − β( n P j=1 sj)2], with α, β ∈ R++. The aggregator is α(s1, ..., sn) = n P j=1

sj. The payoff function becomes

π(si, t) = c(e − si) + nsi P j=1 sj [α n P j=1 sj− β( n P j=1 sj)2].

3.4 Results Cross-partial derivative Submodular games

The analysis is performed for simplicity with 2 players. Since all examples fall within the subgroup of aggregative quasi-submodular games, it is expected that the cross-partial derivative is negative.

For the first submodular example, the Cournot oligopoly with differentiated substitute products, the cost function does not influence the sign of ∂2Π1

∂p1∂p2. Therefore we do not

choose a cost function. The second derivative of the profit curve is: ∂2Π1

∂p1∂p2

= βθ(θ − 1)(qβ₁+ q₂β)θ−2β2qβ−1₁ q₂β−1+ βθ(θ − 1)(θ − 2)(qβ₁ + q₂β)θ−3β2qβ−1₁ q₂2β−16 It is not immediately possible to state whether the sign of the derivative is positive or negative. Given the parameter restrictions, with the aid of Matlab we know that the slope is always negative. This result is in accordance with the fact that the Cournot

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oligopoly with differentiated substitute products is an example of a submodular game. The cross partial derivative of the second submodular example is given by:

∂2_{log Π} 1 ∂p1∂p2 = −k( β β−1) 2_p 1 β−1 1 p 1 β−1 2 (p β β−1 1 + p β β−1 1 )2 7

The second submodular demand function is almost the same as the last supermodular demand function if we choose y = (βθ)1−βθ1 _{and r = β/β − 1. Only Schipper(2004)}

multiplies the denominator with a power term k = _1−βθ1−θ .

Whether the denominator of the demand function is multiplied with the power term k or not, a positive/negative β determines if the goods are substitutes or complements, respectively. But this does not directly influences the sign of the cross partial deriva-tive. The fundamental decision is the choice of cost function. A unit cost function, independent of whether the β is positive or negative (and therefore if r is negative or positive) will always lead to a positive cross partial derivative, which was already verified in section 3.2. An almost similar conclusion holds for a convex cost function, the sign of β does not influence the submodularity property. Any value of β in combination with a convex cost function will lead to a negative cross partial derivative. Note that k must be positive to have a negative cross partial derivative, but this follows automatically from the parameter restrictions.

Here it is stressed out once more that one can not draw the conclusion from the de-mand function or the kind of goods (substitutes of complements) whether or not the game is supermodular. It is all about the specific combination of demand and cost func-tion plus the parameter values which determines the supermodularity/submodularity property.

For the Common-Pool Resource game the slope of the profit function is: ∂2Π1

∂s1∂s2

= 1 − β8

Schipper (2004) stated that β > 0, but this does not give much information about the sign of the second order condition. For 0 < β < 1 the second order condition is positive and for β > 1 the second order condition is negative. Since the second order derivative is not completely as expected (if we assume that games with strategic substitutes are captured by submodular games), we still can look at the slope of the reaction function. Recall that: ∂r(a i) ∂aj = −( ∂ 2_π i ∂ai∂aj )/ ∂ 2_π i (∂ai)2 7

See section 8.2, p.50 (Appendix) for the derivation

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In this example this gives:

∂r(s2)

∂s1

= − 1 − β

2β − 2 = −0.5 Hence the slope of the reaction curve is constantly negative.

All findings are almost completely in correspondence with the fact that aggregative quasi-submodular games lead to negative cross-partial derivatives and therefore downward sloping reaction curves. Only for the Common-Pool Resource game the negative cross-partial derivative only holds for certain parameter values.

4 Evolutionary dynamics for a n-player game

Before we investigate the evolutionary dynamics for different examples of supermodular an submodular games, we first state all the theory behind this matter. In the theory it is assumed that agents set a certain strategy level, note that this can be a price, quantity, investment etc. depending on the kind of game.

4.1 Reaction Curves and Equilibria

In section 3.2 and 3.4 all examples ’behave’ as expected. Since our main goal is to inves-tigate the model in a dynamic environment, we need to find a explicit reaction curves, the equilibrium points and their corresponding stability condition. We first stick to a ’simple’ 2-player game to find a reaction curve for all examples. Later we will extend the model to a n-player game once we introduce the time variable in our story. The rational-ity behind this manner is that if an explicit reaction curve is not found for 2 players, it is almost impossible that it is found for n players. And if it the best response correspon-dence is found for a 2-player game it is more likely it can be extended to a n-player game. To find an equilibrium, the slope of a first order condition is no longer interesting but the first order condition itself. Equating the first order derivative to zero gives rise to the reaction curve.

Since all games of which the evolutionary dynamics are examined are symmetric, the equilibria will also be symmetric. The equilibria are found by finding the fixed points of the reaction curves. The fixed point is the intersection between the reaction curve and the 45◦ line. For example when there are two agents involved, a fixed point (p1, p2) is

p2 = R2(R1(p2)), (note that p1 = R1(p2)) where R1 and R2 are the reaction curves of

firm 1 and firm 2 respectively. In these fixed points equilibrium prices are the same. A explicit reaction curve is not crucial to find an equilibrium but it is essential for further analysis.

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4.2 Learning rules

In a dynamic environment, the reaction curve can be viewed as a best reply to the expected strategies set by the opponent.

p1 = R(pe_2,t) p2 = R(pe_1,t) Frequently used learning rules are

• Naive; the predictions of the producers equals the last observed price pe

t = pt−1

• Nash; pe

t = p∗ producers set the price equal to the Nash equilibrium

• Imitation; players mimics the action of the most profitable player

For the Nash heuristic the player must have a lot of information about the game, there-fore it is logic the cost of the Nash heuristic is higher than the cost of the naive heuristic. The naive heuristic is free, the player just sets it’s price equal to the last observed price value. The imitation heuristic is also free, the player just observes which heuristic was the most profitable in the previous period and mimics that one.

4.3 Heterogeneity in the behaviour of the players

In order to examine the model with heterogeneity in the learning rules we make the assumption that a fixed fraction ρ ∈ [0, 1] uses a certain heuristic, and the remaining fraction 1 − ρ uses another heuristic. Each time period can be viewed as a game in which the agents have one opportunity to set their price. The n agents are drawn from a larger population, and the distribution of all possible combinations within this group of agents follows a binomial distribution with parameter values n and ρ.

One of the interactions that we examine is the interaction between Naive and Nash players, consider that a fraction ρ sets the price equal to p∗ (Nash equilibrium) and the other fraction 1 − ρ reacts to the average price played in the previous iteration. Than we obtain pt = R((n − 1)¯pt−1). Here ¯pt−1 is the average price set over all firms

in the previous period and by applying the law of large numbers can be formalized as ¯

pt−1= ρp∗+ (1 − ρ)pt−1. Putting this together gives:

pt= R((n − 1)(ρp∗+ (1 − ρ)pt−1)) (4)

The equilibrium p∗ is a fixed point of the first difference equation (4).

Intuitively it is maybe strange that for e.g. a Bertrand game with differentiated substi-tutes or complements it is assumed that all players follow the Nash or naive heuristic since differentiation means the process of distinguishing a product and in equation (4) it is lumped together. But one could view the differentiation in something like marketing

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and not so much in elements which leads to price differentiation. So we assume that the differentiation does not lead to difference in the prices.

The dynamics of the equilibrium p∗are related to the eigenvalues of the Jacobian Matrix of the system evaluated at p∗. Specifically, if the eigenvalues all have a magnitude less than one, then the point is an attractor, but if any eigenvalue has a magnitude greater than one, then the point is unstable.

For a fixed fraction there is only one first difference equation (4) and therefore is it un-necessary to calculate the eigenvalues. It just boils down to whether the absolute value of the derivative of equation (4) evaluated at the equilibrium is smaller than one. Thus if ∂[R((n−1)(ρp∗+(1−ρ)pt−1))] ∂pt−1 p∗

< 1, than p∗ is locally stable.

4.4 Evolutionary Dynamics

In the previous section we made the assumption that a fixed fraction uses a certain heuristic. Now we relax this assumption by letting the fraction vary over time which implies that firms can switch from one heuristic to another. What influences the switch is how a certain heuristic performs. This performance can be measured by how much profit a certain heuristic yields. This profit can not be measured directly, but can be approximated by the expected profit.

Assume that there are two heuristic and we pick n firms from a larger population and within this population a fraction ρ follows heuristic 1. Then when a certain firm follows heuristic 1, this means that it is possible that 0 . . . n − 1 firm(s) follow the other heuristic (heuristic 2). Let p1 and p2 be the corresponding prices set by firms using heuristic 1

and heuristic 2, respectively. The expected profit for those agents who follow heuristic 1 can be expressed as follows:

Π1 = F (p1, p2, ρ) = n−1 X k=0 n − 1 k ρk(1 − ρ)(n−1−k)((p1− c1)D((k + 1)p1+ (n − k − 1)p2)) (5) Here constant marginal costs are assumed. For the firms who follow heuristic 2, the expected profits are given by Π2 = F (p2, p1, 1 − ρ).

Given the two profit functions, we create a function which depends on the profit differ-ential, (G(Π1,t−1− Π2,t−1)). The function G(.) shows the evolutionary updating which

means how many firms switch between the heuristics. A convenient function for this evolutionary process is the so-called logit dynamics, which was introduced by Brock and Hommes (1997):

ρt= G(Π1,t−1− Π2,t−1) =

1

1 + exp[−β(Π1,t−1− Π2,t−1− k)]

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Here β is the evolutionary pressure parameter. The higher the chosen value of β the more influence the profits generated by the heuristics have. The parameter k represents the per period information cost that has to be paid for the use of a certain heuristic. One can imagine that e.g. the Nash heuristic is more expensive than the Naive heuristcic. Combining (4) and (6) the evolutionary model can be written as:

pt= R((n − 1)(ρt−1p∗+ (1 − ρt−1)pt−1)) (7)

ρt=

1

1 + exp[−β(Π1,t−1− Π2,t−1− k)]

(8) The equilibrium of this evolutionary model is (p∗, ρk), where ρk = _{1+exp[−βk]}1 since at

the equilibrium it holds that Π1 = Π2.

The equilibrium is locally stable if the eigenvalues of the Jacobian Matrix of equation (7) and (8) evaluated at (p∗, ρk) are smaller than one.

Since three of the four games of which the dynamics are examined fall within the ag-gregative game group, we do expect for these games that the stability conditions depend on n.

5 Local Dynamics for Supermodular Games

In this section we study the local dynamic behaviour of those examples from section 3.1 which gave an explicit reaction curve and a corresponding fixed point by means of numerical simulations.

5.1 Bertrand Oligopoly with differentiated substitute products

5.1.1 Reaction Curve and Equilibria

In contrast to most other supermodular game examples the CES demand function with a unit cost function does give a explicit reaction curve. For n = 2 the payoff function of this game is given by:

Π1 =

ypr−1₁ pr

1+ pr2

(p1− c1)

The first order condition for this game is: ∂ log Π1 ∂p1 = 1 p1− c1 +y(r − 1) p1 − rp r−1 1 pr₁+ pr₂ = 0 9

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Since there are no value restrictions for the parameter y, for simplicity y is set equal to 1. The parameter r should be negative and to make further calculations easy, it is set to -1. The corresponding first order condition then becomes:

∂ log Π1 ∂p1 = 1 p1− c1 − 2 p1 + p −2 1 p−1₁ + p−1₂ = 0

This gives rise to the following reaction curve/best response correspondence for firm 1: p1= R(p2) = c1±

p

c1(c1+ p2)10 (9)

And subsequently for firm 2:

p2 = R(p1) = c2±

p

c2(c2+ p1) (10)

Setting the marginal cost for both firms equal to 1, gives the following plot of the four reaction curves.

Figure 2: The reaction curves for player 1 and 2 of the CES demand function with a unit cost function for y = 1, r = −1, c1 = 1 and c2= 1

The intersections of the reaction curves give are the following four equilibria: (0,0), (-0.618,1.618), (1.1618,-0.618) and (3,3). Since prices can not be negative the relevant equilibria are (0,0) and (3,3).

Before we look at the interaction between the learning rules, we first expand the game to n firms.

The payoff function for firm n looks as follows: Πn= ypr−1_n P jprj (pn− cn) 10

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Setting the parameter r equal to -1 and the marginal cost equal to 1 we obtain the following best response correspondence:

pn= 1 ± q (P j6=np −1 j )2+ P j6=np −1 j −P j6=np −1 j 11

Simplifying the reaction curve gives the following: pn= 1 ±

s

1 + (X

j6=n

p−1_j )−1 ₍₁₁₎

One can verify that the same reaction curve is obtained for 2 firms.

5.1.2 Fixed fraction: Naive expectations versus Nash expectations

We start by finding the Bertrand Nash equilibrium p∗. The equilibrium is the solution of the following equation:

p∗= 1 ± s

1 + p

∗

(n − 1)

The solution to the above equation is p∗ = 2n−1_n−1 12 for both the upward and downward sloping reaction curves. When R((n − 1)p∗) = 1 −q1 +_(n−1)p∗ there is another solution besides p∗ = 2n−1_n−1 namely p∗= 0.

Consider that a fraction ρ ’plays’ p∗ and the other fraction 1 − ρ reacts to the average price ’played’ in the previous period. This gives the following first difference equation:

pt= R(pt−1) = 1 ± r 1 + ((n − 1)(ρ(2n − 1 n − 1) −1_{+ (1 − ρ)p} t−1−1))−1 (12)

The Nash equilibrium p∗ is a solution of this first difference equation and is stable if |∂R(pt−1)

∂pt−1 |p

∗ < 1. The derivative of equation (12) with respect to p_t−1 evaluated at p∗ is

given by: ∂R(pt−1) ∂pt−1 = A ∗ B ∗ C where • A= 1 2(1 + (2n−1) (n−1)2) −1 2 • B = −1((n−1)_2n−12)−2 • C = −(n − 1)(1 − ρ)(2n−1 n−1) −2 12_{This result was found with the aid of Matlab}

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Note that the derivative depends on the number of firms within the game, which is not unexpected since this game is an example of an aggregative game.

It is not easily obtained whether the absolute value of the derivative of equation (12) is smaller than 1, therefore |∂R(pt−1)

∂pt−1 |p∗ is plotted. For the value of ρ, zero is chosen, since

this value gives the highest possible value of the derivative. If the derivative is smaller than 1 for ρ = 0, the derivative will be smaller than 1 for any value of ρ.

Figure 3: The derivative of equation (12) evaluated at p∗

Number of firms n

Figure 2 displays that the derivative is always smaller than 1 and has an horizontal asymptote at y = 0, which means that the equilibrium is always stable independent of the number of firms. These findings are in correspondence with what is found below, the price time paths converges immediately to the equilibrium price of 2.5 and 2 for n = 3 and n = 400, respectively.

(a) n = 3 (b) n = 400

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5.1.3 Evolutionary dynamics for Naive versus Nash firms The profit function of heuristic 1 is: Π1 = p

−2 1 n P j=1 p−1_j

(p1− c1). For the evolutionary model

we need the average profit which is approximated by the expected profit. The expected profit for heuristic 1 given by:

Π1 = F (p1, p2, ρ) = n−1 X k=0 n − 1 k ρk(1 − ρ)(n−1−k)( p −2 1 (k + 1)p−1₁ + (n − k − 1)p−1₂ (p1− c1)) The expected profit for those who follow heuristic 2 is given by Π2 = F (p2, p1, 1 − ρ).

Let heuristic 1 be the Nash heuristic and let heuristic 2 be the naive heuristic and let ρt be the fraction of firms who follow heuristic 1. Then the expected profit function for

the Nash heuristic becomes:

ΠN ash= F (p∗, pt−1, ρt−1) = n−1 X k=0 ρk_t−1(1 − ρt−1)(n−1−k) n − 1 k ( p ∗−2 (k + 1)p∗−1_{+ (n − k − 1)p}−1 t−1 (p1− c1)))

And the expected profit for the naive heuristic is ΠN aive = F (pt−1, p∗, 1 − ρt−1). The

evolutionary process is given by:

ρt= G(ΠN ash,t−1− ΠN aive,t−1− k) =

1

1 + exp[−β(ΠN ash,t−1− ΠN aive,t−1− k)]

(13) The dynamics of the price value are:

pt= R((n − 1)(ρt−1p∗+ (1 − ρt−1)pt−1)) (14)

We know that the reaction function of the CES demand function with a unit cost function looks as follows: pn= 1 ± s (1 + (X j6=n p−1_j )−1) (15)

Combining (14) and (15) gives:

pt= 1 ± r 1 + ((n − 1)(ρt−1( 2n − 1 n − 1 ) −1_{+ (1 − ρ} t−1)pt−1−1))−1 (16)

The full evolutionary model is given by equation (13) and (16). The equilibrium of this model is p∗ = 2n−1_n−1, p∗= 0 and ρk = _1+exp[βk]1 .

To evaluate the stability of p∗, ρk we need to find the Jacobian matrix and the

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since the profit differential contains the binomial distribution. To bypass differentiat-ing the binomial distribution with respect to pt−1and ρt−1, we apply the following steps:

The expected profit function contains the the profit differential which can be rewrit-ten as follows: ∆Π = n−1 X k=0 Ak(ρt−1)Bk(pt−1, p∗)

here Ak(ρt−1) = n−1_k ρkt−1(1 − ρt−1)n−1−k which is independent of pt−1 and

Bk(pt−1, p∗) = D((k + 1)p∗+ (n − 1 − k)pt−1)(p∗− c) − D((k)p∗+ (n − k)pt−1)(pt−1− c)

which is independent of ρt−1. Note that k is the number of Nash firms, in the first case

there is at least 1 Nash firm, therefore D(.) depends on (k + 1) and in the second case the minimum is 0 Nash firms, therefore D(.) depends on k.

To find the Jacobian we take the derivative of the profit differential with respect to ρt−1

and pt−1. Since Ak(ρt−1) is independent of pt−1 and Bk(pt−1, p∗) is independent of ρt−1

there are no cross terms. We obtain: ∂∆Π ∂pt−1 = n−1 X k=0 Ak(ρt−1) ∂Bk(pt−1, p∗) ∂pt−1 and ∂∆Π ∂ρt−1 = n−1 X k=0 ∂Ak(ρt−1) ∂ρt−1 Bk(pt−1, p∗)

The derivatives are evaluated at (p∗, ρk) this will lead to Bk(p∗, p∗) = 0 and therefore ∂∆Π

∂ρt−1 = 0. And |

∂Bk(pt−1,p∗)

∂pt−1 |p

∗ equals zero since the first order condition at p∗ for a

profit maximization problem is zero. Eventually this implies that _∂p∂∆Π

t−1 = 0.

Then the derivatives of the evolutionary process evaluated at the equilibrium are given by: ∂G(∆ΠN ash,t−1− ΠN aive,t−1− k) ∂pt− 1 = ∂G(−k) ∂pt− 1 ∂∆Π ∂pt− 1 and ∂G(∆ΠN ash,t−1− ΠN aive,t−1− k) ∂ρt− 1 = ∂G(−k) ∂ρt− 1 ∂∆Π ∂ρt− 1 Since _∂p∂∆Π t−1 = 0 and ∂∆Π ∂ρt−1 = 0 is 0, it follows that ∂G(∆ΠN ash,t−1−ΠN aive,t−1−k) ∂t−1 and ∂G(∆ΠN ash,t−1−ΠN aive,t−1−k)

∂ρt−1 are equal to zero.

Now it is left to find the partial derivatives of the reaction curve (16) with respect to pt−1 and ρt−1 evaluated at the equilibrium. We obtain

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• ∂R(pt−1) ∂pt−1 |p ∗_,ρ k = A ∗ B ∗ C • ∂R(pt−1) ∂ρt−1 |p ∗_,ρ k = A ∗ B ∗ ((n − 1)( 1 p∗ −_p1∗)) = 0 where • A= 1 2(1 + (2n−1) (n−1)2) −1 2 • B = −1((n−1)_2n−12)−2 • C = −(n − 1)(1 − ρ_k)(2n−1_n−1)−2

Subsequently the Jacobian Matrix evaluated at (x∗, ρk) becomes:

JB=A ∗ B ∗ C 0

0 0

The characteristic equation of any Jacobian Matrix is given by: det(J B − λI), which needs to be solved in order to find the eigenvalues. The characteristic equation of the above Jacobian Matrix is: ((A ∗ B ∗ C) − λ) − λ = −λ(A ∗ B ∗ C) + λ2 = 0 and this implies that λ = (A ∗ B ∗ C) or λ = 0. Both eigenvalues need to be smaller than 1, and therefore we focus on λ = (A ∗ B ∗ C). In the previous section it was already shown that independent of the value of ρk and n, the eigenvalue λ = A ∗ B ∗ C < 1 , and therefore

the dynamics are always stable.

Note that in table 1 and table 2 (see section 8.5 (Appendix)) the dynamics are only shown for the following reaction curve:

pt = 1 +

q

+1 + ((n − 1)(ρt−1(2n−1_n−1)−1+ (1 − ρt−1)pt−1−1))−1 and not for pt = 1 −

q

1 + ((n − 1)(ρt−1(2n−1_n−1)−1+ (1 − ρt−1)pt−1−1))−1. This is because the CES demand

function in combination with a unit cost function should be an example of a supermod-ular game and the later one is a decreasing reaction curve which does not correspond with supermodularity. There is only one disadvantage of only considering the upward sloping reaction curve, namely the downward sloping reaction curve leads to another fixed point (0,0). So we lose one fixed point if we drop the downward sloping reaction curve. Since we have to make a decision we do consider only the upward sloping reaction curve, otherwise it would no longer be an example of a supermodular game.

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(a) The Nash heuristic share time path (b) The price time path

Figure 5: The dynamics for n = 3

Figure 4b shows the price path for n = 3 and table 1 (see Appendix) depicts the dynam-ics of the price value for a different number of firms. Both the figure and the table show that all price paths, independent of the number of firms converges towards their equi-librium. This is in correspondence with the eigenvalues that are found. The eigenvalue, independent of ρk= _{1+exp[−βk]}1 , is always smaller than 1. So increasing the information

cost k only effects ρk but is has hardly any effect on λ and therefore has also no effect

on the price dynamics. Increasing the information cost can not destabilize the price dynamics.

Table 2 (see Appendix) depict the dynamics of the Nash heuristic share for a different number of firms and figure 4a displays the oscillating time series of the Nash players share for n = 3. Both the table and the figure do not show any convergence for the Nash share. For n = 3 one would expect that ρ0 converges to 0.5 since k = 0 but instead

the Nash share jumps between 1 and 0.1972. At some time iterations the Nash share is very low and intuitively you would expect that this would destabilize the situation, but it seems to have no effect on the convergence of the price path.

Unfortunately it is not possible to plot a bifurcation diagram, because the profit dif-ferential could not be simplified and therefore the simulation were executed in Matlab. But since the dynamics are always stable, the bifurcation diagram would not be that interesting and probably shows an upward sloping straight line.

5.2 The Diamond-type Search model with a quadratic cost function

5.2.1 Reaction Curve and Equilibria

The Diamond-type Search model is introduced as a model with payoff function:

Πn = αx(n)Pm6=nx(m) − C(x(n)) where C(.) can be any cost function. We have

chosen for the cost function C(x(n)) = x(n)3, than the profit function looks as follows: Πn= αx(n)

P

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by:

α X

m6=n

x(m) − 3x(n)2 (17)

Equating this equation to zero, for n = 2 the reaction curve is given by: x(1) =r α

3x(2)

Figure 6: The Reaction curves of the Diamond-type Search model for trader 1 and 2 with α = 4 with C(x(1)) = x(1)3

In this plot is clearly visible that there are two equilibria, namely x∗ = 0 and x∗ = α₃ = 4₃. From equation (17) we can the derive the reaction curve for a world with n traders and is given by: x(n) =s α 3 X n6=m x(m) (18)

The fixed points of equation (18) are the solutions of the following equation: x∗=r α

3(n − 1)x

∗

which gives x∗ = 0 and x∗ = α₃(n − 1).

Here the interaction between between the Nash and naive expectation rule for a fixed fraction is examined. Consider that a fraction ρ set their effort level equal to the Nash

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equilibrium and the remaining investors set their effort value equal as the best response to the effort level in the previous period. Which gives the following first difference equation: xt= r α 3(n − 1)(ρx ∗_{+ (1 − ρ)x} t−1) (19)

This equilibrium is stable when the derivative of equation (19) with respect to xt−1

evaluated at the equilibrium is smaller than 1. The derivative of equation (19) is: 1 2 1 p(α 3(n − 1)(ρx∗+ (1 − ρ)xt−1)) α 3(n − 1)(1 − ρ) (20)

Evaluating the derivative at x∗ = α₃(n − 1) gives: 1 2 1 p(α 3(n − 1) α 3(n − 1)) α 3(n − 1)(1 − ρ) = 1 2(1 − ρ)

For stability |1₂(1 − ρ)| < 1, which always hold since 0 ≤ ρ ≤ 1. This implies that for every value of ρ and n the equilibrium α₃(n − 1) is stable.

Evaluating equation (20) at x∗= 0 will given an outcome of zero which is always smaller than one. Therefore x∗ = 0 is always stable.

(a) Bifurcation diagram for x∗= α₃(n − 1) (b) Effort time path for n = 100

Figure 7: Diamond-type Search Model with cost function x(n)3 for x∗ = α₃(n − 1), α = 1 and k = 0

Figure 7 depicts the dynamics when x∗ = α₃(n − 1) is chosen as equilibrium point. The bifurcation diagram (figure 7a) shows that there always is one stable equilibrium for 2 < n < 10 and no other dynamics are obtained. Figure 7b indicates that the effort path converges towards the equilibrium for n = 100. Both figures are in correspondence with the analysis stated above the figures.

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(a) n = 100 and ρ = 0.9 (b) n = 100 and ρ = 0.5

(c) n = 10 and ρ = 0.9 (d) n = 10 and ρ = 0.5

Figure 8: Diamond-type Search Model with cost function x(n)3 for ρ = 0.5, α = 1 and x∗ = 0

Figure 8 shows the dynamics when the equilibrium point x∗ = 0 is chosen. It is depicted in the figures that the dynamics do converge bot not towards x∗ = 0. The higher the value of ρ, the higher the value of the convergence point. In general it holds that the dynamics converge towards α₃(n − 1)(1 − ρ). For figure 9 (a),(b),(c) en (d) the dynamics converge to 3.3, 16.5, 0.3 and 1.5 respectively. This equilibrium point was not found analytically, but only by simulation. But from the analysis it was expected that the dynamics are always stable independent of n and ρ, and this is in correspondence with the dynamics found.

5.2.3 Evolutionary dynamics for Naive versus Nash traders

The profit function of firm 1 in a market with n firms is Π1 = αx(1)P_m6=nx(m) − x(1)3.

For the evolutionary dynamics, we need the average profits for both heuristic which can be approximated by the expected profits. The expected profit for the investors who follow heuristic one is given by:

Π1 = n−1 X k=0 n − 1 k ρk(1 − ρ)n−1−k[αx(1)[(k + 1)x(1) + (n − 1 − k)x(2)] − x(1)3]

This expected profit function can be simplified by applying a trick: we know that the expectation of the binomial distribution is given by (n − 1)ρ.

By applying the trick, the simplified expected profit function is given by: Π1 = αx(1)[((n−

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The expected profit for the firms who follow heuristic 2 can be found in a similar way and is given by:

Π2 = F (p2, p1, 1 − ρ) = αx(2)[((n − 1)ρ + 1)x(2) + (n − ((n − 1)ρ) − 1)x(1)] − x(2)3

Subtracting the expected profits gives:

Π1− Π2 = α((n − 1)ρ + 1)x(1)2− x(1)3− α((n − 1)ρ + 1)x(2)2+ x(2)3

= α((n − 1)ρ + 1)(x(1)2− x(2)2) + (x(2)3− x(1)3) The evolutionary process is given by:

ρt=

1

1 + exp[−β(Π1,t−1− Π2,t−1− k)]

(21) Let heuristic 1 be the Nash heuristic and let heuristic 2 be the naive heuristic and let ρt

be the fraction of firms who follow the Nash heuristic. Than the evolutionary process is given by:

ρt=

1

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= 1

1 + exp[−β(α((n − 1)ρt−1+ 1)(x∗2− x2t−1) + (x3t−1− x∗3) − k)]

(23) The dynamics of the effort value are:

pt= R((n − 1)(ρt−1x∗+ (1 − ρt−1)xt−1)) (24)

The reaction function of the Diamond-type Search model with as cost function x(n)3 looks as follows: xt= s α 3 X m6=n x(m) (25)

Combining (24) and (25) gives xt=

r α

3((n − 1) ∗ (ρt−1x

∗_{+ (1 − ρ}

t−1)xt−1)) (26)

Than the complete full evolutionary model is given by equation (23) and (26). The equilibrium of this model is x∗= 0, x∗ = α₃(n − 1) and ρk = _1+exp[βk]1 .

To figure out the stability condition, the Jacobian Matrix of equation (23) and (26) is derived and displayed below:

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JB=a ∗ 1 2 α 3(n − 1)(1 − ρt−1) a ∗ 1 2 α 3((n − 1)(x ∗_{− x} t−1)) b ∗ c b ∗ d x ∗_,ρ k where • a = _√ 1 α 3((n−1)(ρt−1x ∗_+(1−ρ t−1)xt−1)) • b = − exp[−β(α((n−1)ρt−1+1)(x ∗2_−x2 t−1)+(x3t−1−x∗3)−k)] (1+exp[−β(α((n−1)ρt−1+1)(x∗2−x2t−1)+(x3t−1−x∗3)−k)])2 • c = −β((−α((n − 1)ρt−1+ 1)2xt−1) + 3x2t−1) • d = −β(α(n − 1))(x∗2− x2 t−1)

The Jacobian Matrix evaluated at (x∗ = 0, ρk) becomes:

JB= 0 0 a ∗ b a ∗ c where • a = _{(1+exp[−β(−k)])}− exp[−β(−k)]2 • b = 0 • c = 0

All elements of the Jacobian matrix evaluated at x∗ = 0 are zero, which implies that the eigenvalues are zero. Since the eigenvalues are zero, the equilibrium (x∗ = 0, ρk) is

always stable.

The Jacobian Matrix evaluated at (x∗ = α₃(n − 1), ρk) becomes:

JB= ₁ 2(1 − ρk) 0 a ∗ b a ∗ c where • a = − exp[−β(−k)] (1+exp[−β(−k)])2 • b = −β((−α((n − 1)ρt−1+ 1)2(α₃(n − 1))) + 3(α₃(n − 1))2) • c = 0

The characteristic equation is given by: (1₂(1 − ρk) − λ) − λ = 0, which implies that λ = 0

and λ = 1₂(1 − ρk). Since ρk varies between 0 and 1, both eigenvalues are smaller than

1. Therefore the equilibrium (x∗= α₃(n − 1), ρk) is always stable. So we have found that

both equilibrium points are always stable.

Like in the fixed fraction case, an upward sloping straight line is observed as bifurcation diagram. Remarkable is that, the dynamics which are evaluated at x∗ = 0 as equilibrium point do not converge to zero but to the other equilibrium α₃(n − 1). So in general, all dynamics converge towards α₃(n − 1) and are always stable, independent of n.

Increasing the information cost have absolutely no effect on the dynamics, they stay stable.

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5.2.4 Evolutionary dynamics for Naive versus Nash versus Imitation traders In this section the previous section is expanded with a extra rule, namely the imitation heuristic rule. The imitators follow that rule (Nash or naive) which is the most prof-itable.

The theory behind the evolutionary dynamics for three heuristic is the same as for two heuristics. In a similar way the full evolutionary model can be derived, only the nonlinear dynamical system is no longer two-dimensional but a three-dimensional system. There are two evolutionary updating functions since there exist three heuristic rules. The first function (G(ΠN ash− ΠN aive)) gives ρ1, which is the fraction of investors who follow the

Nash heuristic and the second function (F (ΠImitation− ΠN aive)) gives ρ2 which is the

fraction of imitators. These two function imply that 1 − ρ1− ρ2 is the fraction of naive

players.

From the preceding section we know that in general the payoff differential for two heuris-tic rules is given by:

Π1− Π2 = α((n − 1)ρ + 1)(x(1)2− x(2)2) + (x(2)3− x(1)3)

Since there are three heurisitics the payoff differential functions are:

ΠN ash− ΠN aive = α((n − 1)ρ + 1)(x∗2− x2t−1) + (x3t−1− x

∗3

)

ΠImitation− ΠN aive= α((n − 1)ρ + 1)(x2Imitation− x2t−1) + (x3t−1− x3Imitation)

Here xImitation can be xt−1 or x∗ in each time iteration.

The dynamics for the effort value are given in the following formula:

xt= R((n − 1)(ρ1,t−1x∗+ ρ2,t−1xt−1+ (1 − ρ1,t−1− ρ2,t−1)xt−1)) (27)

The reaction curve of the Diamond-type Search model is given by: x(n) =s α

3 X

m6=n

x(m) (28)

Combining the previous two equations gives: xt=

r α

3((n − 1)(ρ1,t−1x

∗_{+ ρ}

2,t−1xImitation+ (1 − ρ1,t−1− ρ2,t−1)xt−1)) (29)

The evolution of the effort value xt, and the evolution of the fraction of Nash players and

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nonlinear dynamical system: xt= r α 3((n − 1)(ρ1,t−1x ∗_{+ ρ} 2,t−1xImitation+ (1 − ρ1,t−1− ρ2,t−1)xt−1)) (30) ρ1,t = 1 1 + exp[−β(α((n − 1)ρ1,t−1+ 1)(x∗2− x2_t−1) + (x3_t−1− x∗3) − k)] (31) ρ2,t = 1

1 + exp[−β(α((n − 1)ρ2,t−1+ 1)(x2_Imitation− x2_t−1) + (x3_t−1− x3_Imitation) − k)]

(32) To figure out the stability of the dynamics, we need to derive the Jacobian matrix. The Jacobian Matrix is a 3 × 3 matrix where the first column shows the derivative with respect to xt−1, the second column shows the derivative with respect to ρ1,t−1 and the

third column shows the derivative with respect to ρ2,t−1. The Jacobian Matrix of

equa-tion (30), (31) and (31) evaluated at x∗, ρ1,k and ρ2,k is given by:

JB=    1 √_α 3(n−1)x ∗ 1 2 α 3(n − 1)(1 − ρ1,k− ρ2,k) 1 √_α 3(n−1)x ∗ ∗ 0 1 √_α 3(n−1)∗x ∗ ∗ 0 y ∗ −β((−α((n − 1)ρ1,k + 1) − 2xt−1) + 3x2t−1 y ∗ 0 0 y ∗ −β((−α((n − 1)ρ2,k + 1) − 2xt−1) + 3x2t−1 0 y ∗ 0    where • y = _{(1+exp[−β(−k)])}− exp[−β(−k)]2

If x∗ = 0, again all elements of the Jacobian Matrix are equal to zero and the corre-sponding eigenvalues will be zero. This implies that the dynamics when x∗ = 0 are always stable.

When x∗ = α₃(n − 1), the characteristic equation is (√α 1

3(n−1)∗ α 3(n−1) 1 2 α 3(n − 1)(1 − ρ1,k−

ρ2,k) − λ)λ2 = 0 which can be simplified to 1₂(1 − ρ1,k− ρ2,k) − λ)λ2 = 0. This implies

that λ = 0 and λ = 1₂(1 − ρ1,k − ρ2,k). But then again 0 ≤ (1 − ρ1,k− ρ2,k) ≤ 1 and

therefore both λ0s are always smaller than 1.

Overall, including an extra heuristic gives exactly the same results as in the previous section. Both equilibrium points are always stable and the dynamics always converge to the equilibrium point x∗ = α₃(n − 1) and not towards the other equilibrium point.

6 Local Dynamics for Submodular game

Here we study the local dynamic behaviour of the submodular examples from section 3.3 which gave an explicit reaction curve and corresponding equilibrium points.

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6.1 Bertrand Oligopoly with differentiated complementary products

6.1.1 Reaction curve and Equilibria

The payoff function of the CES demand function in combination with a convex cost function for n = 2 is given by: Π1= p1∗ D1− D12where D1 = (βθ)

1 1−βθ p 1 β−1 1 (p β β−1 1 +p β β−1 2 )k (k = 1−θ

1−θβ) with parameter restrictions 0 < θ < 1, θ < 1 and β < 0.

For convenience the log function is used which eventually lead to a first order condition given by: 1 p1 − 1 β−1 p1 + k β β−1p 1 β−1 1 p β β−1 1 + p β β−1 2 = 013

We have chosen the following parameter values: β = −0.5, θ = −0.5 and it follows that k = _1−θβ1−θ = 2. This eventually gives the following best-response curve for player 1:

p1 =

−125 343 p2

For this model the parameter values need to be chosen very carefully. If one would choose β = −1 and θ = −0.5, which would also satisfy the parameter restriction, this would give the reaction curve: p1 = 1₄p2. Which is an upward sloping reaction curve and

is strange for a submodular game. In general the reaction curve is: p1= (_{(−kβ)(β−2)}β−2 )p2.

The slope coefficient of this reaction curve is not for all β < 0 negative. This is in contrast to what Vives (2000) suggest that for every β < 0 we have an example of a submodular game.

Figure 9: Reaction curves Bertrand game with differentiated complementary products β = −0.5 and θ = −0.5

The figure shows that for 2 investors, there is one fixed point namely (0,0).

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The payoff function for investor i in a game with n investors looks as follows: Πi = pi∗ Di− D2i where Di = (βθ) 1 1−βθ p 1 β−1 i (P jp β β−1 j )k . The first order condition is given by:

1 pi − 1 β−1 pi +k β β−1p 1 β−1 i P jp β β−1 j = 0

Substituting the parameter values β = −0.5 and θ = −0.5 and k = 2, will eventually gives the following best-response curve for player i:

pi= −125 343 ( X j6=i p 1 3 j) 314 ₍₃₃₎

The fixed point of this equation is p∗ = 0.

Suppose that a fraction ρ sets the price equal to p∗ and the other fraction 1 − ρ uses as price expectation the price of the previous period. Given equation (33) we obtain the following: pt= R((n − 1)(ρp∗+ (1 − ρ)pt−1)) (34) = −125 343 [(n − 1)(ρp ∗1/3_{+ (1 − ρ)p}1/3 t−1)]3 (35)

The equilibrium p∗ is a fixed point of the first difference equation.

The derivative of the first difference equation (35) with respect to pt−1 is given by:

∂R((n − 1)(ρp∗+ (1 − ρ)pt−1)) ∂pt−1 = −125 343 3[(n − 1)(ρp ∗1/3 + (1 − ρ)p1/3_t−1)]2(n − 1)(1 − ρ)1 3p −2 3 t−1

Evaluating the derivative at the fixed point p∗ = 0 will give an outcome which is equal to zero, which is always smaller than 1. Therefore the equilibrium point is always stable, independent of ρ and n. Remarkable here is that the derivative of the first difference equation (35) does depend on n, which is expected since this is an example of an aggregative game. But once it is evaluated at the fixed point, the variable n drops out. This is suprising since we would expect that the stability condition did depend on n because this game falls within the group of aggregative games.

14

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6.1.3 Evolutionary dynamics for Naive versus Nash firms The profit function of firm 1 in market with n firms is given by: Π1 = p1(βθ 1 1−βθ p 1 β−1 1 (P jp β β−1)k j ) − (βθ1−βθ1 p 1 β−1 1 (P jp β β−1)k j )2. Filling in β = −0.5, θ = −0.5 and k = 2 yields the next profit function:

Π1 = p1(0.25 1 0.75 p −2 3 1 (P jp 1 3 j)2 ) − (0.250.751 p −2 3 1 (P jp 1 3 j)2 )2

To examine the evolutionary dynamics, the expected profit function is necessary. Con-sider a market with n firms and two heuristic, than the expected profit function for those firm who follow heuristic 1 looks as follows:

Π1 = F (p1, p2, ρ) = n−1 X k=0 n − 1 k ρk(1 − ρ)(n−1−k)∗ Z where Z = (p1(0.25 1 0.75 p −2 3 1 ((k+1)p 1 3 1+(n−k−1)p 1 3 2)2 ) − (0.250.751 p −2 3 1 ((k+1)p 1 3 1+(n−k−1)p 1 3 2)2 )2).

In a similar way the expected profit function for the firms which use heuristic 2 can be derived and can be written as Π2 = F (p2, p1, 1 − ρ).

Since the interaction between the Nash heuristic and naive heuristic is analyzed, we set heuristic 1 equal to the Nash heuristic and heuristic 2 is equal to the naive heuristic. The expected profit function for the Nash heurstic looks as follows:

ΠN ash= F (p∗, pt−1, ρt−1) = n−1 X k=0 n − 1 k ρk_t−1(1 − ρt−1)(n−1−k)∗ Y where Y = (p∗(0.250.751 p ∗−2₃ ((k+1)p∗13+(n−k−1)p 1 3 t−1)2 )−(0.250.751 p −2 3 1 ((k+1)p∗13+(n−k−1)p 1 3 t−1)2 )2). The expected profit function for the naive heuristic is ΠN aive= F (pt−1, p∗, 1−ρt−1). Knowing

the two profit function, the profit differential can be calculated and the evolutionary process is given by:

ρt= G(ΠN ash,t−1− ΠN aive,t−1− k) =

1

(36) Besides the dynamics of the Nash share heuristic, the dynamics of the price are also of interest and these are given by:

On the stability of Nash Equilibria in super(sub)-modular games : an evolutionary perspective

On the Stability of Nash Equilibria in Super(Sub)-Modular

Games: an Evolutionary Perspective

Contents

1

Introduction

2

An introduction to Supermodular and Submodular games

3

Examples of Supermodular and Submodular game

4

Evolutionary dynamics for a n-player game

5

Local Dynamics for Supermodular Games

6

Local Dynamics for Submodular game