Eric Langlais
email: langlais@univ-nancy2.fr.

Asymmetric information, self-serving bias and the pretrial negotiation impasse

Keywords: JEL classification: D81, K42 litigation, pretrial bargaining, self-serving bias, risk aversion

.

Introduction

The fact that informational imperfections introduce biases in litigants' assessment of the outcome of trial, thus entailing efficiency losses in a dispute between two parties, is beyond debate. The issue which is still controversial in the literature is what causes the disagreement between parties leading to the pretrial bargaining impasse?

In the hunt for the most powerful theory of litigations, two prominent theories have been suggested. According to the "optimistic approach" [START_REF] Priest | The selection of disputes for litigation[END_REF], [START_REF] Shavell | Suits, settlement and trial: a theoretical analysis under alternative methods for the allocation of legal costs[END_REF]), the failure of pretrial negotiations occurs because the plaintiff is more confident than the defendant about his own chances to win at trial -the more confident the plaintiff relative to the defendant, the more likely the trial. As a result, the optimistic model predicts that the subset of cases going to trials is not a representative sample of all tried cases, but it is biased towards those for which the plaintiff has a trial win rate close to 50% [START_REF] Priest | The selection of disputes for litigation[END_REF], [START_REF] Waldfogel | The selection hypothesis and the relationship between trial and plaintiff victory[END_REF][START_REF] Waldfogel | Reconciling asymmetric information and divergent expectation theories of litigation[END_REF]). In the "strategic approach", the existence of information asymmetries between the parties explains why disputes are sometimes inefficiently solved in front of Courts. In a seminal paper, [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF] assumes that the defendant has the private information concerning whether he was negligent, and shows that cases going to trial are those for which the plaintiff has the best chances to prevail1 .

The available empirical evidence is mixed. [START_REF] Priest | The selection of disputes for litigation[END_REF] and [START_REF] Waldfogel | The selection hypothesis and the relationship between trial and plaintiff victory[END_REF] find evidence in favour of the prediction of a 50% prevailing rate for plaintiffs, but Hughes and Snyder (1985) and [START_REF] Katz | Measuring the demand for litigation: is the English rule really cheaper?[END_REF] conclude that it is more likely to be less than 20%. [START_REF] Waldfogel | Reconciling asymmetric information and divergent expectation theories of litigation[END_REF] rejects the assumption that information asymmetries exist for cases going to trial, while [START_REF] Osborne | Who should be worried about asymmetric information in litigation?[END_REF] finds evidence that they do exist, but interestingly not in a way supporting [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF]'s assumption that defendants' degree of fault is a private information. He suggests that this is actually more likely to be connected with differences between parties' predictive power of the outcome at trial. In other words, litigants are neither equally skilled nor have the same ability to assess their chances to prevail at trial. Hence, we come back to the question of what explains that litigants are unequally skilled in predicting the outcome at trial? [START_REF] Farmer | Pretrial bargaining with self-serving bias and asymmetric information[END_REF] and [START_REF] Bar-Gill | The success and survival of cautious optimism: legal rules and endogenous perceptions in pre-trial settlement negotiations[END_REF],2005) investigate a promising line of research.

They assume that litigants exhibit the self-serving bias, a form of bounded rationality that has been documented in the experimental literature. [START_REF] Akerlof | The economic consequence of cognitive dissonance[END_REF] early discussed the influence of cognitive limits and self-manipulation of believes for economic decisions. [START_REF] Farber | Why is there disagreement in bargaining[END_REF] long ago argued that neither divergent expectations nor asymmetric informations provide sufficient explanations for the existence of a disagreement in bargaining; in contrast, the existence of cognitive limits, and various forms of bounded rationality, provide the most powerful arguments.

More recently, a growing literature in the area of behavioral law and economics has also provided empirical and experimental evidence for the presence of cognitive limits such as anchoring effects and optimistic or self-serving bias on behalf of individuals in civil litigations. There exist convincing proofs that cognitive biases are exhibited by both well experienced lawyers and judges [START_REF] Ichino | Are Judges Biased by Labor Market Conditions?[END_REF], [START_REF] Marinescu | Are Judges Sensitive to Economic Conditions? Evidence from UK employment tribunals[END_REF], [START_REF] Rachlinski | Heuristics and Biases in Specialized Judges : The Case of Bankruptcy Judges[END_REF]Wistrich (2007), Viscusi (2001)) and more naive individuals [START_REF] Babcock | Explaining bargaining impasse: the role of self-serving biases[END_REF]). For example, Babcock and Loewenstein survey experimental studies showing that people display the self-serving bias, that is, a systematic tendency for individuals to interpret facts in civil litigations in a way which is favorable to themselves: given identical facts, an individual given the role of a plaintiff (defendant) will interpret those facts as more favorable for the plaintiff (respectively for the defendant).

In their paper, [START_REF] Farmer | Pretrial bargaining with self-serving bias and asymmetric information[END_REF] introduce the self-serving bias in a model of litigations à la [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF] and analyse its incidence on the trial rate and the well-being of parties (i.e. the amount for which they settle the case). Typically, the authors2 find that an increase in the optimistic bias of the informed party (the defendant) has an ambiguous effect on the frequency of trial and settlement demand. In contrast, an increase in plaintiff's self-serving bias (uninformed party) increases the incidence of trial but decreases his settlement proposal. [START_REF] Bar-Gill | The success and survival of cautious optimism: legal rules and endogenous perceptions in pre-trial settlement negotiations[END_REF],2005) argues that in an evolutionist context (with symmetric information), the factors allowing a specific kind of optimistic litigants to survive are twofold: on the one hand, optimism works as a commitment device in the bargaining process, allowing them to appear as tougher negotiators settling only for more favorable terms -this pushes litigants towards a high degree of optimism. On the other hand, optimism as a general feature of litigants, makes less likely that all the cases being filed end up in a settlement -hence, this effect pushes litigants towards less optimism. The tension between these two forces leads to an equilibrium where the (homogenous) population of litigants share the same level of "cautious optimism".

Our paper departs from both works. We consider a heterogenous population of optimistically biased litigants, and we assume that the source of the asymmetric information is the size of the plaintiff's bias. In this framework, we show that during the pretrial bargaining round, the population of plaintiffs is split according to their type between those (more optimistically biased) going to trial and those (less biased) accepting the defendant's settlement offer. We show that changes in the optimistic bias of the uninformed litigant (defendant) have unambiguous consequences which are similar to the predictions of the "naïve" optimistic approach. We also study various shifts in bias distribution for the informed party (plaintiff), showing that additive shifts have predictable consequences consistent with the optimistic approach, while multiplicative ones entail more ambiguous consequences.

Finally, the paper introduces another channel for probability manipulation, which is distinct from the optimistic bias. Experimental economics also provides lots of proofs that naïve, as well as experienced individuals display a typical pattern of probabilities transformation, termed "probabilistic risk aversion". [START_REF] Decidue | On the Intuition of Rank-Dependent Utility[END_REF] as [START_REF] Weber | Reasons for Rank-Dependent Utility Evaluation[END_REF] present several arguments allowing to rationalize such a probability transformation process. This kind of deliberate mental accounting is the way in which individuals take into account and minimize the loss, disappointment or pain they will suffer when making an error in the assessment of the outcome associated with their decision. In a sense, probabilistic risk aversion is a behavioral theory explaining the degree of confidence of an individual regarding his priors and his ability to take the best decision; while the self-serving bias is supported by a theory of self-manipulation of beliefs reflecting that, even if they have the true and objective information about the situation, individuals are prone (more or less deliberately) to reinterpret the facts in a way favorable to themselves. For practical purpose, the point is that individuals display simultaneously both attitudes. In this paper, we introduce the characterization provided by [START_REF] Yaari | The Dual Theory of choice under risk[END_REF] for the (probabilistic) risk aversion assumption, and show that changes in risk aversion of the informed litigant have an ambiguous effect on negotiations and trials, but typically different from those resulting from the optimistic bias.

The paper is structured as follows. Section 2 presents the main assumptions of our screening model of litigation and its equilibrium. Section 3 discusses the comparative statics results. Section 4 briefly considers the influence of probabilistic risk aversion. Section 5 concludes.

The model with the informed Plaintiff

We will modify the screening model of [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF] in the following manner.

assumptions and timing of the model

We consider a plaintiff which is harmed by an accident that may be the result of negligence or wrongdoing by another party, the defendant. The loss suffered by the plaintiff in case of accident is D > 0 and corresponds to the damages awarded by the Court in case the trial is in favour of the victim. The compensation D is public information. We denote p the probability that the judgment at trial be in favour of the plaintiff.

We assume that the (risk-neutral) plaintiff displays a self-serving bias (σ > 1), and thus has an assessment of the prior corresponding to his chances to win at trial denoted σ.p, which is larger than his true probability: in other words, the plaintiff interprets the facts of the case as more favorable for himself than they really are from an objective point of view. His assessment of the risky prospect faced at trial X = (D -C p , p; -C p , 1p), writes:

E(X) ≡ σpD -C p
with C p > 0 his legal costs. However, the prior σp is private information for the plaintiff and is not observable by the defendant: the defendant only knows that the plaintiff's bias is a random variable σ ∈ [a, b] (with a > 1) distributed according to a probability function characterized by the cumulative function G(σ) and the density g(σ), which are public information. In what follows, σ is labelled as the type of the plaintiff3 . In order to rule out secondary difficulties, we introduce the following assumption:

Assumption 1: the hazard rate G g is increasing.

Finally, we consider that the (risk-neutral) defendant also displays a self-serving bias denoted by σ d < 1, such that his assessment of the plaintiff's chances to win at trial is σ d .p which is supposed to be public information; this implies that when facing the risky prospect of trial Y = (-(D + C d ), p; -C d , 1p), the expected loss borne by the defendant at trial is:

E(-Y ) ≡ σ d pD + C d with C d > 0 his legal costs.
The pretrial bargaining process has two main stages 4 , following Nature's choice of the type of the plaintiff σ in [a, b], and also following the plaintiff having filed his case:

-In a first stage, the defendant makes a "take-it-or-leave-it" offer to the plaintiff, denoted s, in order to reach a settlement of the case.

-In the second stage, depending on his type, the plaintiff accepts the offer (thus, the case is settled) or rejects it, in which case the parties go to trial.

Finally we assume that pD > C ≡ C p + C d meaning that the case to be solved is socially valuable. Remark that this implies that: apD -C p > 0, meaning that the weakest plaintiff's type (who believes that the defendant has the best chances to be seen as not liable by the Court) always has an incentive to go to trial.

the equilibrium

The (Bayesian) equilibrium is described in terms of the amount for which the parties settle, s (the equilibrium offer of the defendant to the plaintiff), and of the probability of a trial corresponding to the marginal plaintiff σ(s), the one who is indifferent between accepting the offer or rejecting it and going to trial.

In the second stage, the plaintiff σ chooses between accepting the offer which gives him a sure gain s, and going to risky trial X in which case the expected payoff obtained is σpD -C p . As a result, the plaintiff σ accepts the offer s as soon as: s ≥ σpD -C p . Otherwise, he rejects it. The marginal plaintiff σ(s) is thus defined by the condition:

σ(s)pD -C p = s (1)
Any plaintiff having a prior more pessimistic than the marginal plaintiff (any σ ≤ σ(s)) will also accept the offer, while any plaintiff having a more optimistic prior (σ > σ(s)) will go to trial.

Going back to the first stage, we can write the loss function according to which the defendant will set his best offer. With probability G (σ(s)), the defendant knows he will face a plaintiff prone to accept his offer, and thus will incur the cost s to settle the case. But with probability 1 -G (σ(s)), the defendant knows he will face a plaintiff more optimistic than the marginal one, and thus will have to pay the expected cost σ d pD + C d . The defendant will announce the best offer ŝ ≥ 0, which minimize the loss function:

L (s) = G (σ(s)) × s + (1 -G (σ(s))) × (σ d pD + C d ) (2)
under condition (1).

Proposition 1: In an interior equilibrium, the offer ŝ and the marginal plaintiff σ are set according to the following conditions:

ŝ = σpD -C p (3) G g |(σ) = σ d -σ + C pD (4)
such that the probability of a trial is π = 1 -G(σ).

Proof: If ŝ > 0 and σ ∈]a, b[ are an admissible interior solution for the minimization of (2), then the First Order Condition writes as:

G (σ) -g (σ) 1 pD ((σ d p -σp)D + C p + C d ) = 0 (5)
Rearranging the various terms leads to (4). Note that the Right Hand Side in (4) must be positive since the gains of the negotiation must be positive for an interior solution to exist:

σ d - σ + C pD > 0 ⇔ ŝ < σ d pD + C d .
Existence may be proven as in [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF]. Moreover, under assumption 1, the Left Hand Side in ( 4) is increasing with σ; the RHS is decreasing with σ. Thus, the solution for (3)-( 4) is unique and satisfies the second order condition (which requires that L (ŝ) > 0).

The first LHS term in (5) is the marginal cost of the defendant's offer: raising this offer leads to an increase in the loss incurred by the defendant which is equal to the probability of settlement. The second LHS term in (5) is the marginal benefit of the offer which may be split in two components:

-on the one hand, the effect of raising the offer on the probability of trial (which decreases),

d ds (1 -G(σ(s)) = -g (σ) 1
pD < 0; this term reflects the efficiency of the screening of the various plaintiff's types due to an increase in the settlement offer. As the defendant raises his offer, some (types of) plaintiffs abandon the trial and prefer to settle their case. But remark that the higher the expected judgment, the lower the impact of the offer on the probability of trial, i.e. the higher the expected damage obtained by the plaintiff at trial in case of victory, the weaker the efficiency of the screening role played by the settlement offer.

-on the other hand, the gains of the negotiation for the marginal plaintiff, (σ d -σ)pD + C > 0, since the amicable settlement of the case allows the defendant to save his judiciary costs and extract those of the plaintiff plus the value of the judgment. This second term obviously reflects the gain associated with the screening of the plaintiffs according to their type. Remark that as a result of the assumption on the domain of definition of individual bias, we have: σ d -σ < 0, and thus the larger the self-serving biases (i.e. the larger the disagreement between parties), the smaller the gains of the negotiation.

Remark that this framework shares some features with both the optimistic approach and the strategic one. The condition σ d -σ < 0 implies that the gains of the negotiation (at equilibrium) are smaller than the transaction costs at trial. Were the parties both risk neutral and having no self-serving bias, these gains would be reduced to the aggregate transaction costs of a trial:

C p + C d .
But given that the parties do not have the same perception of the risk of a trial (they have different priors) the negotiation gains are different from the sum of transaction costs of a trial:

E(Y ) -ŝ = C p + C d ⇔ σ d p -σp = 0.
The effect highligthed in the "optimistic approach" of litigations [START_REF] Priest | The selection of disputes for litigation[END_REF], [START_REF] Shavell | Suits, settlement and trial: a theoretical analysis under alternative methods for the allocation of legal costs[END_REF]) is that litigants displaying an exogenous optimistic prior will be biased in favor of a trial since they both overestimate their own chance to prevail in the Court. In our set-up and despite the existence of this optimistic bias, some plaintiffs may prefer to settle their case. The result is in fact consistent with the well known intuition of strategic models, according to which pretrial negotiations may fail even when the gains of the negotiations are positive, due to the existence of an asymmetric information. Here, the population of plaintiffs is split between those going to trial because the gains of the negotiation are smaller than their expected payment in case of judgment, and those accepting the defendant's offer for the opposite reason.

Comparative statics

We first investigate the influence of the expected damage and transaction costs. Then, we focus on the role of both individual self-serving biases. We make an extensive use of the fact that the effect of a given shock on the equilibrium may be understood thanks to its specific impact on the marginal benefit of the settlement offer, as previously defined.

influence of expected damages and fee shifting

This analysis is easy to perform since as remarked in the proof of proposition 1, G g |σ in the LHS term in ( 4) is an increasing function of σ while σ dσ + C pD in the RHS term of ( 4) is decreasing function of σ.

The first issue of interest is the impact on the equilibrium of a increase in pD the expected damage at trial. Remark first that this may be the result of an increase in p or in D, and in fact both have the same influence:

Proposition 2: An increase in pD: I) decreases the marginal plaintiff (hence the probability of trial increases).

II

) has an ambiguous effect on the equilibrium offer.

Proof. I) It is easy to verify that an increase in pD has a negative impact on the RHS in (4).

Hence σ decreases with pD. II) An increase in pD has two effects on ŝ in (3): a direct effect which is positive, and an indirect effect through the impact on the marginal plaintiff, which is negative; hence the ambiguity.

The ambiguity associated with the settlement offer reflects the existence of two opposite forces.

On the one hand, there is a direct and positive effect: given that the increase in pD improve the plaintiff expected payoff at trial, the defendant must increase his settlement offer (all else held equal) in order to convince more plaintiffs to accept it; but on the other hand, there is an indirect and negative effect, given that the borderline plaintiff type decreases. Specifically, this decrease in the marginal type of plaintiff is explained by the decrease in the marginal benefit of an offer (see the discussion following proposition 1) with pD, through both a contraction of the gains of the negotiation and a reduction of the impact (decrease in the efficiency) of the settlement offer on the frequency of trials. Thus, the increase in the expected damage at trial has a clear-cut and positive effect on the trial rate.

Another issue close to the previous one, is the influence of the stakes at trial, as a result of the choice of a specific fee-shifting rule. We obtain the next result: Proposition 3. The rate of trial is the smallest under the American rule, the largest under the English rule, and intermediate under the Continental rule.

Proof. Consider a general formulation of the fee-shifting rule, such as the French or Continental rule, where the judge has the discretionary power to transfer to the loosing party a part of the winner's costs (called "depens", such as taxes, expertise expenditures, but excluding attorney's fees): Let α ∈ [0, 1] be the proportion of the defendant's costs borne by the plaintiff when he looses at trial. Similarly, let β ∈ [0, 1] be the proportion of the plaintiff's costs borne by the defendant when the plaintiff wins at trial. The American rule where each party simply bears its own costs is obtained when α = β = 0. The British rule, according to which the party loosing at trial has to bear the aggregate costs of the trial, is the case where α = β = 1. It can be shown that for any α and β (being public information), the equivalent to condition (4) writes:

G g |(σ) = σ d -σ + C p(D + βC p + αC d )
where the larger C θ+βC p +αC d , the higher σ and the smaller 1 -G(σ). All else held equal, we have:

C pD > C p (D + βC p + αC d ) > C p (D + C p + C d )
Thus, under the American rule the borderline type is larger than under the French rule, which one is larger also than under the British rule.

It may also be verified that the impact of fee-shifting on the settlement offer is ambiguous.

It is obvious that the channel through which the value of the stakes at trial influences the equilibrium is the same as for the expected judgment pD: the larger D + βC p + αC d , the smaller the marginal benefit of the settlement offer. As remarked before as the value of the stakes at trial increases, it implies both a contraction of the gains of the negotiation and a reduction of the impact (decrease in the efficiency) of the settlement offer on the frequency of trials. Thus, it has a positive effect on the trial rate.

In this sense, proposition 3 predicts that the American rule is the best promoter of settlements. This is a well known result in the literature whatever the informational context, including the "naïve optimistic" model (with purely exogenous optimistic beliefs: [START_REF] Priest | The selection of disputes for litigation[END_REF]Klein (1984), Shavell (1982)) and the strategic approach (Bebchuck (1984), [START_REF] Farmer | Pretrial bargaining with self-serving bias and asymmetric information[END_REF]), but including also alternative analysis of the litigation system such as the rent-seeking (Farmer and Pecorino (199), [START_REF] Katz | Measuring the demand for litigation: is the English rule really cheaper?[END_REF]) or the auction-theoretic approach [START_REF] Baye | Comparative analysis of litigation systems: an auction-theoretic approach[END_REF]).

It has also some connections with the evolutionist-based analysis recently developed. [START_REF] Bar-Gill | The success and survival of cautious optimism: legal rules and endogenous perceptions in pre-trial settlement negotiations[END_REF],2005) shows that the "cautious optimistic" type supporting an Evolutionary Equilibrium, is higher under the American rule as compared to the British one. However5 , we do not verify the prediction of the author who observed that "since a greater level of optimism increases the likelihood of a bargaining impasse, [the usually recognized] advantage of the American rule is reduced [in a context where the degree of optimism is endogenous]". Our result in contrast shows that for a heterogenous population of litigants, this advantage does not vanish: the tendency of the British rule to induce more legal expenditures is enhanced by the optimism of litigants.

changes in individual biases

The next results focus on the effects attached to the change of individual priors, and thus in the perception of the risk at trial experienced by each litigant, beginning with: Proposition 4 is consistent with the intuition of Farmer and Pecorino (2002, p 164) that the specific way in which the optimistic bias of a party affects the frequency of trials at equilibrium may depend on the nature of the information asymmetry and/or on the order of play between the parties: the ambiguity concerning the role of σ d obtained by [START_REF] Farmer | Pretrial bargaining with self-serving bias and asymmetric information[END_REF] disappears when the plaintiff is supposed to have private information. This is fully explained in our framework as follows: the marginal benefit of the settlement offer (and specifically, the gain of the negotiation, since the impact of the offer on the rate of trial does not depend on σ d ) increases

with the defendant's bias, yielding fewer trials, and thus a higher settlement offer.

On the other hand, a rise in σ d means that the defendant becomes less optimistic (given that σ d < 1): as σ d increases, the bias regarding his perception of the chances that the plaintiff prevails is reduced, and his own assessment of the likelihood of winning becomes closer to the true probability.

Hence, the result that fewer trials occur is exactly the one more usually obtained in the "naïve optimistic" model [START_REF] Priest | The selection of disputes for litigation[END_REF], [START_REF] Waldfogel | Reconciling asymmetric information and divergent expectation theories of litigation[END_REF]).

The next two propositions highlight how alternative changes in the relevant domain for the value of plaintiff's self-serving bias affect the equilibrium:

Proposition 5: An additive shift to the right in the range of plaintiff's types: I) implies a less than proportional increase in the marginal type;

II) increases the probability of trial;

III) increases the equilibrium offer.

Proof. We define (see also [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF]) an additive shift to the right of the range of plaintiff's types as a t-translation of plaintiff's types, such that σ is now distributed on the interval [a + t, b + t] (with t 0) with the cumulative Γ(σ) and the density γ(σ) functions satisfying the following correspondences with the primitives G(σ) and g(σ):

Γ(σ) = G(σ -t) γ(σ) = g(σ -t)
In fact, these two conditions characterize a family of distribution functions which is parametrized by t 0, where t = 0 gives us the primitives, and t > 0 leads to a distribution with a higher mean type but having identical higher order moments. In this case, the condition (4) may be substituted with the general formulation:

G g |(σ-t) = σ d -σ + C pD (6)
with π = 1 -G(σt) and ŝ given by (3). I) Differentiating ( 6) gives:

dσ dt = 1 Ω G g |(σ-t) with: Ω ≡ G g |(σ-t)
+ 1 > 0 according to the second order condition. Moreover, under assumption

2: G g |(σ-t)
> 0. Thus, it is obvious that dσ dt > 0 but dσ dt < 1. II) As a result, π = 1 -G(σt) increases with t. III) Given that the marginal type increases with t, it is also straightforward to see that the equilibrium offer ŝ = σpD -C p also increases with t.

Note that the impact on the settlement offer is related only to the (direct) effect on the borderline type: as the support of the distribution of σt shifts to the right (with t), the gains of the negotiation are uniformly reduced for all the population of plaintiffs6 , which leads to an increase in the borderline type and a decrease in the likelihood of settlement. Thus, as the plaintiffs appear as (uniformly) tougher negotiators, the defendant must accept an increase in the settlement offer in order to convince some of them to accept it. Proposition 4 gives predictions that are close to the findings by [START_REF] Farmer | Pretrial bargaining with self-serving bias and asymmetric information[END_REF] in the case of additive biases. This of course reflects the very definition of an additive shift to the right of the range of plaintiff's types since such a shift increases the mean type and leaves unchanged the higher order moments. It is also worth noticing that the results in proposition 4 are consistent with those more usually obtained in the "naïve optimistic" model. However, changing our definition of the shift in the possible range of plaintiff's optimistic bias will introduce new effects in equilibrium: Proposition 6. A mean-preserving proportional shift in the range of plaintiff's types: I) decreases the marginal type if σ < µ; otherwise, the effect is ambiguous; II) has an ambiguous effect on the probability of trial; III) decreases (increases) the equilibrium offer if the marginal type decreases (respectively, increases).

Proof. We define (see also [START_REF] Bebchuk | Litigation and settlement under imperfect information[END_REF]) a mean-preserving proportional shift of the distribution of plaintiff's types as the composition of an additive shift to the left (a µ(1t)-translation, with t 1 and µ = E(σ)) plus a multiplicative shift of plaintiff's types, such that σ is now distributed on the interval [ta + µ(1t), tb + µ(1t)] with a cumulative Γ(σ) and a density γ(σ) satisfying the following correspondences with the primitives G(σ) and g(σ):

Γ(σ) = G σ -µ t + µ γ(σ) = 1 t g σ -µ t + µ
Once more, these two conditions characterize a family of distribution functions which is parametrized by t 1, where t = 1 gives us the primitives, and t > 1 gives us a new distribution with the same mean µ = E(σ) but which is more spread than the primitive distribution; thus it has moments of higher orders which are larger than those of the primitive. In this case, condition (4) may be now substituted with the general formulation:

G g |( σ-µ t +µ) = 1 t σ d -σ + C pD (7)
with π = 1 -G σ-µ t + µ . I) Differentiating (7) gives:

dσ dt = 1 Ω × G g |( σ-µ t +µ) × σ -µ t - G g |( σ-µ t +µ)
It is obvious that σ < µ ⇒ dσ dt < 0 although if σ > µ then dσ dt has an ambiguous sign. II) Similarly, π may decrease or increase with t since:

dπ dt = -g σ -µ t + µ × 1 t × dσ dt - σ -µ t
Hence the result. III) Given the ambiguity on the marginal type, it is also straightforward to see that the equilibrium offer ŝ = σpD -C p may as well increase (if the marginal type increases) as decrease (respectively if the marginal type decreases) with t.

In contrast with the uniform perturbation of the distribution, the spread of all possible types appears to have a more complex, thus less easy to predict impact on the pretrial negotiations. Once more, the impact on the settlement offer reflects the (direct) effect of the shift t on the borderline type. But this last effect is now ambiguous: on the one hand, the efficiency of the separation obtained through the settlement offer is inversely related 7 to t, while the relationship between t and the gains of the negotiation appears here as ambiguous. The potential benefit obtained through the negotiation by the marginal plaintiff depends on his specific range among all possible types, and thus in this sense, it depends on some specific features of the concentration of the distribution (such as the concentration around the mean type) 8 .

Proposition 6 confirms the ambiguous results of [START_REF] Farmer | Pretrial bargaining with self-serving bias and asymmetric information[END_REF] in the case of a multiplicative optimistic bias, which contrast with those predicted by the "naïve optimistic model". To see this, let us remind that [START_REF] Priest | The selection of disputes for litigation[END_REF] and [START_REF] Waldfogel | Reconciling asymmetric information and divergent expectation theories of litigation[END_REF] showed that when the errors made by the litigants in predicting 9 the outcome at trial increase, then the likelihood of a trial also increases: this is because the chances are higher that plaintiff's optimistic estimate of prevailing at trial are larger than defendant's ones. Now, let us interpret the meanpreserving proportional shift in the range of plaintiff's types as representing, from the defendant's point of view, the expectation of an increasing variability in plaintiff's priors on the outcome at trial. In this sense, proposition 6 means that as the plaintiff's priors are expected to become more spread, the impact of the equilibrium become less clear, depending for example on some additional characteristics of the distribution of types 10 .

Risk aversion

We briefly sketch now how the results extend to the case where the plaintiff has a self-serving bias and is risk averse in the sense of Yaari.

Assume that the plaintiff has preferences which satisfy the axiomatic of Yaari's model [START_REF] Yaari | The Dual Theory of choice under risk[END_REF]); thus, there exists a probability transformation (or probability weighting) function ϕ(p)

which is endowed with the basic properties that ϕ : [0, 1] → [0, 1] is unique, continuous and increasing in p, with ϕ(0) = 0 and ϕ(1) = 1. We will assume that ϕ(p) is (at least twice)

8 Note that the bracketed term in the RHS of (7) also writes:

σ d -σ-µ t + µ + C pD + 1-t t (σ -µ) + µ ;
hence as t increases, the RHS in (7) may increase or decrease according to the sign of σµ, yielding accordingly a decrease or an increase in the equilibrium rate of trials 9 In the interpretation of the optimistic model suggested by [START_REF] Priest | The selection of disputes for litigation[END_REF], litigants perform unbiased estimates, or rational expectations, for their chances of prevaling at trial. In contrast and by definition, we consider here the situation where litigants have biased estimates of their chances to win at trial. 1 0 Remark that dσ dt may equivalently be written as:

dσ dt = 1 Ω × G g | σ-µ t +µ
× (e -1)

where e = G g σ-µ t G g
is the (partial) elasticity of the rate of hazard with respect to σ-µ t , evaluated at σ-µ t + µ . Hence, the results may depend on the concentration of the distribution, as captured by e.

differentiable, with ∀p ∈ [0, 1] : ϕ (p) > 0; accordingly, the plaintiff is risk averse [START_REF] Yaari | The Dual Theory of choice under risk[END_REF].

In this setting, the satisfaction level or "anticipated utility" of the plaintiff σ facing a risky prospect at trial X = (D -C p , p; -C p , 1p), writes:

E ϕop (X) ≡ (1 -ϕ(σp))(-C p ) + ϕ(σp)(D -C p ) = ϕ(σp)D -C p
In other words, E ϕop (X) is the subjectively transformed expected outcome of the prospect, since the probability of each outcome is replaced by a subjective weight of likelihood. Specifically, according to the convexity of ϕ(.), the plaintiff places a weight of likelihood to the worst (best) outcome which is larger (smaller) than its prior: i.e. 1ϕ(σp) > 1σp (respectively ϕ(σp) < σp).

The method in paragraph 2.2 can be performed to solve for the equilibrium of the pretrial negotiation game, and it can be shown that in an interior equilibrium, the offer ŝ and the marginal plaintiff σ are now set according to the following conditions:

ŝ = ϕ(σp)D -C p (8) G g |(σ) = 1 ϕ (σp) σ d - ϕ(σp) p + C pD (9)
The way plaintiff's risk aversion affects the negotiation and the equilibrium rate of trial depends on its impact on the RHS in (9); thus, it is linked to the marginal benefit of the settlement offer which is defined as:

-g(σ) 1 ϕ (σp)pD . ((σ d p -ϕ(σp))D + C)
Note first that the impact of risk aversion on the gains of the negotiation is ambiguous since we may have: σ d pϕ(σp) 0. On the one hand, as previously found for risk-neutral litigants, the existence of the self-serving bias reduces the gains of the negotiation and induces litigants to go more frequently to trial. On the other hand, the risk aversion alone would imply that σp > ϕ(σp)

-this means that if both litigants had had the same prior (σp), then the risk neutral defendant's posterior belief would have been less pessimistic than the risk averse plaintiff's one. Hence, the plaintiff's risk aversion per se tends to increase the negotiation gains, thereby inducing an increase in the borderline type and yielding finally less trials. Thus, as a result of the existence of both the self-serving bias and risk aversion, there is some ambiguity in the relationship between the parties' beliefs (regarding the chances that the plaintiff prevails at trial) and the trial rate: the sign of σ d pϕ(σp) is ambiguous. The plaintiff's risk aversion may dampen the influence of those biases and improve the gains of the negotiation. But the opposite result may also obtain.

Second, risk aversion also affects the equilibrium through the impact (efficiency) of the settlement offer on the probability of trial, as reflected by the term -g(σ)

1 ϕ (σp)pD . On the one hand, if ϕ (σp) < 1, meaning that the plaintiff's risk aversion increases with his prior (the difference σpϕ(σp) increases with σp), then the separation between plaintiffs is easier for the defendant: this leads, all else held equal, to less trials. The opposite results is obtained when ϕ (σp) > 1, meaning that the plaintiff's risk aversion decreases with his prior. Thus, the variation of the plaintiff's risk aversion with its prior introduces another source of ambiguity in the analysis.

The complete comparative static may be developed and it confirms the ambiguous influence of the plaintiff's risk aversion; in contrast, it can also be shown that the results of propositions 2 to 6 still hold 11 under risk aversion. Given that most of the time, the self-serving bias is seen in the literature as an "irrational bias", while the probabilistic risk aversion is described as a "rational" one, our results are quite paradoxical: although the previous analysis suggests that the influence of the former is intuitive and not too troublesome for pretrial negotiations, the impact of the latter seems to be more puzzling.

Conclusion

There is a longlasting debate in experimental economics concerning the relevant interpretation for the growing evidence that people proceed to probabilities transformation or manipulation in a way not consistent with rational inference and Bayesian updating rules. On the one hand, for psychologists this reflects a kind of bounded rationality due to the presence of cognitive dissonance or inconsistency, revealing that people use heuristics rather than sophisticated processes for the assessment of their priors. This paper focuses on the self-serving bias and analyses its consequences on litigations when this is a source of asymmetric information. On the other hand, other researches in social sciences argue that the systematic departure from Bayesian inference exhibited in experimental situations does not necessarily rule out any explanation consistent with the theory of procedural rationality, such as the (probabilistic) risk aversion assumption. These two explanations are more complementary than rival, and they may be reconciled; this paper is a proposal in this sense. Two main conclusions may be drawn. The first one is that, roughly speaking, our results are consistent with those of models including a more naïve assessment of individual beliefs, excepted when we consider non trivial changes in the population of litigants' types. The second one is that the risk aversion assumption seems to have more dramatic consequences than the optimistic bias, to the extent that changes in the former are less predictable than shifts in the latter, which are now documented in the literature.

Finally, to the extent that the self-serving bias may be socially, institutionally or culturally determined, the issue for Law & Economics is not only how the existence of self-serving bias may affect the judicial process, but also how innovations emerging from the legal process (regarding procedural rules as well as substantive law) may modify the optimism of litigants. Focusing on the pretrial impasse, our paper like some few other existing in the literature, gives insights on the first question, but has not paid enough attention to the second one. Future research, both at the theoretical and empirical levels, should examine the kind of changes promoting uniform (additive) rather than erratic (multiplicative) shifts in the optimism of litigants. In our framework, we have seen that these shifts have far different effects; but a limitation of the analysis is that they are supposed to be purely exogenous.

Proposition 4 :

 4 A rise in σ d : I) increases the marginal plaintiff (hence the probability of trial decreases), II) increases the equilibrium offer. Proof. Straightforward: I) the RHS in (4) increases with σ d ; II) σ d has a positive impact on ŝ only through the marginal plaintiff (see (3)).

[START_REF] Daughety | Settlement[END_REF] provides a complete survey of the strategic analysis of litigations.

More precisely, Farmer and Pecorino (2002) study the consequence of both a multiplicative and an additive bias.

As discussed in[START_REF] Bar-Gill | The evolution and persistence of optimism in litigation[END_REF], there is an unavoidable tension between the structure of information and the assumption that parties are not aware of their own bias. In fact, the distinction between p and the priors σ i × p is a harsh simplification; but it is very tractable for the purpose of modelization: it means that there exists some public information which is used by both parties to assess their own priors -the very existence of the self-serving bias implies that people select and interpret the same available information in a way favorable to themselves.[START_REF] Akerlof | The economic consequence of cognitive dissonance[END_REF] long ago discussed the psychological motivations (rationalization process) for such information selection and self-manipulation of beliefs, which allows only the partial and imperfect learning of his own bias by an individual. Experimental economics confirms that such biases are persistent and that de-biasing people is difficult[START_REF] Camerer | Experimental study of law, Chapter 21 in: Handbook of Law and Economics[END_REF]).

Daughety et Reinganum (1994) argue that this is the more relevant assumption. First, Courts impose delays on pretrial bargaining. Then, parties chose to self-restrain the negotiations in order to lower the associated costs.We may also consider that parties often reach an agreement on the steps of the tribunal, just some minutes before the trial.

There is a major difference between our approach and the evolutionsit one: our result allows the separation between plaintiffs according to their optimistic types -the less (more) optimistic prefering to settle their case (the trial, respectively); on the other hand, the EE is consistent with the settlement of all cases.

To see this, remark that the RHS in (6) is equivalent to σ d -(σt) + C pD -t .

The impact of s on the likelihood of a trial is now equal to -1 t g.

1 See our working paper(Langlais,

2008b). For an analysis of asymmetric information on plaintiff 's risk aversion, see also[START_REF] Farmer | Pretrial negotiations with asymmetric information on risk preferences[END_REF] andLanglais (2008a).


Ludovic A Julien

Fabrice Tricou

Market Price Mechanisms and Stackelberg General Equilibria

This paper considers Stackelberg competition in a general equilibrium framework with production. The working of market power and the con-…gurations of strategic interactions are complexi…ed by the presence of an active leader. Two market price mechanisms are here studied: one is associated with the Stackelberg-Walras equilibrium, the other is linked to the Stackelberg-Cournot equilibrium. In the context of an exchange economy with a production sector, several results are obtained about equilibria mergings and about welfare comparisons.

Introduction

The concept of oligopoly equilibrium initially introduced by Stackelberg (1934) has mainly been developed in partial equilibrium analysis (see for instance Anderson and Engers (1992), Friedman (1992) and [START_REF] Tirole | Theory of industrial organization[END_REF]). This paper aims at extending the analysis of oligopolistic competition proposed by Stackelberg to a general equilibrium framework. We therefore propose to analyze the shapes and the consequences of market power under Stackelberg competition for an overall economy.

The concept of Stackelberg equilibrium is here modeled in the context of an exchange economy with a production sector. The simple model we develop is su¢ cient to display the diversity of behaviors and of strategic interactions associated with this complex form of competition. The "Stackelberg-Walras equilibrium" combines perfectly and imperfectly competitive behaviors, whereas the This paper is a piece of a more general research devoted to cooperation failures and coordination failures in market economies. It was presented in the international Workshop "General Equilibrium and Strategic Interactions: Theories and Applications", which held at the University of Paris Ouest-Nanterre in September 2008. We are grateful to the participants for their questions and remarks. A preliminary and incomplete version was discussed in the CoFail seminar. We are grateful to Christian Bidard, Olivier Musy, Antoine Rebeyrol and Aurélien Saidi for their remarks. Any remaining errors are the authors'responsibility.

"Stackelberg-Cournot equilibrium" only involves strategic behaviors. In the former speci…cation, the asymmetry takes place on the same side of the market and also between opposite sides of the market: some agents will act competitively as price-takers, while the other agents shall behave oligopolistically, with one leader and one follower. In the latter speci…cation, only the …rst type of asymmetry remains, the exchange taking place between one leader and followers.

Stackelberg competition is here studied under two di¤erent market price mechanisms. One Stackelberg general equilibrium concept is therefore associated with each market price mechanism. The …rst one comes from a market equilibrium condition, with a market excess demand function for each commodity depending on a price system manipulated by the aggregate strategic supply. In this approach, the equilibrium market prices are determined for any given strategies (Walrasian step) and then the oligopolists'strategies are decided (Cournotian step). The Cournot-Walras equilibrium is the typical concept in this line initially developed by Codognato andGabszewicz (1991), (1993), and later pursued by d'Aspremont et al. (1997) and by [START_REF] Gabszewicz | Oligopoly equilibria in exchange economies[END_REF]. The second market price mechanism is implemented in strategic market games proposed by [START_REF] Shapley | Trade using one commodity as a means of payment[END_REF] and developed by [START_REF] Amir | A strategic market game with complete markets[END_REF]). Without money, the market relative price for each pair of commodities is the inverse ratio of the aggregate strategic supplies. These two kinds of market price mechanisms have been largely used in the literature devoted to general equilibrium under strategic interactions [START_REF] Gabszewicz | Strategic multilateral exchange. General equilibrium with imperfect competition[END_REF] and Giraud ( 2003)).

Tackling any Stackelberg general equilibrium concept, we assume the individual positions and the timing of moves as given, and therefore do not question the way a speci…c agent could or should become a leader [START_REF] Amir | Stackelberg versus Cournot equilibrium[END_REF]). The purpose of this paper is rather the introduction of strategic interactions à la Stackelberg between interrelated markets with production1 . Correlatively, two interesting features can be put forward when casting Stackelberg competition into a general equilibrium framework. First, the market demand addressed to the producers is here made endogenous, which overcomes the lack of microfoundations that occurs with the usual assumption of an exogenous market demand function. Second, the model displays di¤erent kinds of heterogeneity and throws light on their consequences in terms of welfare. It especially integrates asymmetries across markets, which cannot be captured in partial equilibrium analyses 2 .

Three types of results are obtained about mergings between equilibria and about production or welfare comparisons. For instance, when the productivity parameters of both oligopolists take the same value, the Stackelberg equilibria coincide with the Cournot equilibria for the two considered market mechanisms. And when these parameters take di¤erent values, there is no Pareto domination between the Stackelberg equilibria and the Cournot equilibria.

The paper is organized as follows. In section 2, we present the environment and de…ne the economy. In section 3, we study the equilibrium allocation under the Stackelberg-Walras mechanism and establish two propositions. In section 4, we analyze the equilibrium allocation under the Cournot-Stackelberg mechanism and establish two propositions. In section 5, we compare the two kinds of Stackelberg allocations, and we introduce the competitive equilibrium in the general ranking of levels of production. In section 6, we conclude.

The economy

We consider an exchange economy with a production sector. It includes a …nite set H of agents, indexed h, with h = 1; :::; n + 2; and two divisible commodities (1 and 2). As an exchange economy, this framework displays agents 1 and 2 buying good 2 and the n other agents buying good 1 as consumption goods. As an economy with a production sector, this framework diplays agents 1 and 2 buying good 2 as an input, producing good 1 for self-consumption and for sale. The prices of the two goods expressed in a numéraire are denoted p 1 and p 2 ; and we may refer to the relative price p1 p2 as p. The preferences of any agent h 2 H are represented by the following utility function: 1) ln x h2 , with 2 (0; 1), 8h = 1; :::

U h = ln x h1 + (
; n + 2, (1)
where x h1 and x h2 represent the quantities of both goods consumed by any agent h. The structure of the initial endowments is assumed to be the same as in [START_REF] Gabszewicz | Oligopoly equilibria in exchange economies[END_REF]:

! h = (0; 0) , 8h = 1; 2, (2)
! h = 0; 1 n , 8h = 3; :::; n + 2. (3)
The amount of the second good is thus initially given and spread among the agents of the second type, each of them owning by assumption the same quantity 1=n. The …rst good does not exist a priori in the economy, but can be produced by the …rst type of agents, who are endowed with a constant returns to scale technology 3 . Each linear technology tranforms the second commodity into the …rst one, according to the following speci…cations:

y 11 = 1 1 z 12 and y 21 = 1 2 z 22 , with 1 , 2 > 0, (4)
where y 11 and y 21 are the amounts of commodity 1 that agents 1 and 2 can produce when using quantities z 12 and z 22 of good 2. So, the parameters 1 and

The two producing agents always behave strategically. When they compete together à la Stackelberg, the …rst agent is assumed to be the leader and the second is the follower. This asymmetry on the same side of the market could be based on a discrepancy in productivity: assuming 1 < 2 , or

1 1 > 1 2
, would be a way to justify that the more productive agent is granted the leader position, while the less productive agent is granted the follower position. Under Stackelberg or Cournot competition, both oligopolists have two strategic decisions to make: which quantities y 11 and y 21 of good 1 to produce (which determines through (4) the amounts z 12 and z 22 of good 2 to be bought as inputs), and which amounts s 11 and e 21 of good 1 to supply in exchange for good 2 on the market. The strategy sets for the supplies of both oligopolists are respectively S 1 = fs [START_REF] Gabszewicz | Oligopoly equilibrium in pure exchange economies[END_REF] p2 of this good and x 21 = y 21 e 21 of good 1. The n agents endowed in good 2 behave either competitively or strategically. When acting competitively, these agents have only one decision to make: which quantities of the two goods (x h1 ; x h2 ) they want to consume, taking the price system (p 1 ; p 2 ) as given. Denoting E h the strategic set of any agent h, h = 3; :::; n + 2, we have E h = f?g. In this case, the allocation (x h1 ; x h2 ) of agent h satis…es p 1 x h1 + p 2 x h2 p2 n , 8h = 3; :::; n + 2. When acting strategically, they will try to manipulate the price system (p 1 ; p 2 ) in order to obtain a more favorable rate of exchange. Let us denote e h2 the pure strategy of agent h, h = 3; :::; n+2, with e h2 2 0; 1 n . The strategic set of any agent h, h = 3; :::; n+2, is then S h = e h2 2 R + j 0 e h2 1 n . Any h will thus obtain in exchange of e h2 a quantity p2 p1 e h2 of good 1. In that case, the market process leads to the …nal allocation (x h1 ; x h2 ) such that x h1 = p2 p1 e h2 and x h2 = 1 n e h2 . Beyond the three shapes of behavior at stake, two kinds of market price mechanism will be considered. The …rst one is basically a market equilibrium condition, de…ned for given strategies; while the second one insures the turn over of the supplied goods, the relative price being the inverse ratio of brought quantities. We here introduce a leader à la Stackelberg under both market price mechanisms in order to analyze the strategic interactions and to determine the associated equilibrium allocations.

Let us now de…ne an economy for this environment 5 .

De…nition 1 An economy is a collection of agents, endowments, production and strategic sets = f(! 1 ; Y 1 ; S 1 ) ; (! 2 ; Y 2 ; E 2 ) ; (! h ; E h )g h=3;:::;n+2 .

For this economy, we …rst study the Stackelberg-Walras equilibrium (SWE), for which E h = f?g, 8h = 3; :::; n + 2. We then relax the price-taking assumption and study the Stackelberg-Cournot equilibrium (SCE), for which

E h = e h2 2 R + j 0 e h2 1
n , 8h = 3; :::; n + 2. We …nally compare these two equilibria to the Cournot-Walras equilibrium (CWE) and to the Cournot equilibrium (CE).

The Stackelberg-Walras equilibrium

In the SWE framework, it is considered that agents having endowments in good 2 act competitively, whereas the other agents behave strategically. The leader manipulates the follower's decision and all these oligopolists manipulate the price by restricting their supply, while the many other agents are price-takers.

A SWE for the economy is a non cooperative equilibrium of a game where the players are the oligopolists, the strategies are their production and supply decisions and the payo¤s are their utility levels. The Stackelberg-Walras allocations depend on competitive and strategic decisions. This equilibrium concept can be viewed as a subgame perfect equilibrium of a three-stage game 6 . In the …rst step the competitive equilibrium is computed for any value of each strategy. In the second step the follower's reaction functions are determined. In the third step the leader's optimal decisions are made. The story is solved by backward induction, considering …rst all the competitive behaviors, then the decisions of the follower and …nally the choices of the leader.

De…nition 2 A SWE for = f(! 1 ; Y 1 ; S 1 ) ; (! 2 ; Y 2 ; E 2 ) ; (! h ; E h )
Therefore, each agent h, h = 3; :::; n + 2, solves the following maximization program M ax

(x h1 ;x h2 ) ln x h1 + (1 ) ln x h2 s.t. p 1 x h1 + p 2 x h2 p2
n . This leads notably to the vector of competitive demand functions for commodities 1 and 2, i.e. x h (p 1 ; p 2 ) = n p2 p1 ; 1 n , 8h = 3; :::; n + 2. The aggregate demand function in good 1 which is addressed to the oligopolists is thus x 1 = p2 p1 . The marketclearing condition for good 1 then writes p2 p1 = s 11 + e 21 , which leads to:

p 1 p 2 = s 11 + e 21 . (5)
Equation ( 5) veri…es

@ p 1 p 2 @
> 0: the relative price depends on the structure of preferences. And @ p 1 p 2 @(s11+e21) < 0: the oligopolists can get a better price restricting their supplies.

The strategic plan of the follower is determined by two elements: s/he manipulates the market price and s/he takes the leader's strategy as given. The objective function of the follower can be written as the payo¤ function V 2 (e 21 ; y 21 ) = ln (y 21 e 21 ) + ( 1) ln

The optimality conditions with respect to e 21 and y 21 , i.e. @V 2 =@e 21 = 0 and @V 2 =@y 21 = 0, yield:

y 21 e 21 = (1 ) s11 (s11+e21)
From [START_REF] Codognato | Cournot-Walras equilibria in markets with a continuum of traders[END_REF] we can deduce the two reaction functions of the follower:

e 21 (s 11 ) = s 11 + r 2 s 11 , (8)
y 21 (s 11 ) = 2 2 + (1 2 ) r 2 s 11 (1 )s 11 . (9)
Since s 11 is the only leader's decision in ‡uencing the relative price, these two reaction functions depend on s 11 and not on y 11 . These functions inherit the properties of the relations given in (1) and ( 4) and are all continuous. We must have y 21 > e 21 > 0. Indeed, e 21 > 0 is obtained when s 11 < 2 (see below Equation ( 12)), i.e. 2 1 > 2 , which is assumed to be veri…ed. And y 21 > e 21 (which guarantees that

y 21 > 0) if 1 2 + 1 4 2 1 > 1. This inequality stands if and only if ' 1 2 > 0, with ' 1 2 2 1 2 + 1
4 . We easily verify that ' 0 on 1 2 ; 1 . Additionally, @e21 @s11 > 0 for s 11 2 h 0; 4 2 i and @e21 @s11 < 0 for s 11 2 h 4 2 ; 2 i

7 . We also have @ 2 e21 @s 2 11 < 0. The same last three properties hold 7 The equilibrium strategies in [START_REF] Gabszewicz | Strategic multilateral exchange. General equilibrium with imperfect competition[END_REF] will show that s11 2 h for y 21 (s 11 ). Finally, @e21 @ 2 < 0 and @e21 @ > 0: the strategic supply of the follower increases when her/his productivity goes up and depends on the structure of preferences.

The strategic plan of the leader is determined by two elements: the leader manipulates the market price, as given by ( 5), and the follower strategy e 21 . Using the same argument as previously, one obtains the following payo¤ function for the leader: V 1 (s 11 ; y 11 ) = ln (y 11 s 11 ) + ( 1) ln

The optimality conditions with respect to s 11 and y 11 , i.e. @V 2 =@s 11 = 0 and @V 2 =@y 11 = 0, yield:

y 21 e 21 (s 11 ) = (1 ) e21(s11) (s11+e21(s11)) 2 = (1 ) 2 , (11)
where s11+e21(s11) e 21 (s 11 ) 2 y 21 . This gives the vector of equilibrium strategies of the leader: ;

(s 11 ; ỹ11 ) = 4
(1 + ) 4 .

We obviously have (s 11 ; ỹ11 ) 2 R 2 ++ , and ỹ11 > s11 as > 0. We deduce the equilibrium follower's equilibrium strategies ẽ21 (s 11 ) and ỹ21 (s 11 ):

(ẽ 21 ; ỹ21 ) = (2 1 2 ) 4 2 1 ; 2 2 + (1 2 ) 2 1 (1 ) 2 4 2 1 . (13)
From ( 5), ( 12) and ( 13), the equilibrium relative price is:

p1 p 2 = 2 1 . (14)
We have

@ p1 p 2 @ 1 > 0.
As the leader's productivity increases, commodity 1 becomes less valued relatively to commodity 2.

The equilibrium allocations are:

(x 11 ; x12 ) = 2 2 2 2 1 ; (1 ) 4 2 1 , (15)
(x 21 ; x22 ) = 2 2 2 1 2 2 1 2 ; (1 ) 2 1 2 2 1 2 ! , (16)
(x h1 ; xh2 ) = 1 2 1 n ; 1 n , 8h = 3; :::; n + 2. ( 17
)
The corresponding utility levels are:

Ũ1 = ( ) + ln 2 (1 + ) ln 1 2 ln 2, (18)
Ũ2 = ( ) + 2[ln(2 1 2 ) ln 2 ln 1 ] ln 2 , (19)
Ũh = ( 1) ln( 1) + (ln ln 2 ln 1 ) ln n, 8h 6

= 1; 2, (20)
where ( ) (1 + ) ln + ( 1) ln( 1). It can be noticed that the utility of each of the two oligopolists depends positively on his/her productivity and negatively on the other's productivity 8 . The welfare of each competitive agent decreases with their number (through the endowment e¤ect) but increases with the leader's productivity.

It is meaningful to compare the market outcomes holding under the SWE with these that would prevail if both producers played a Cournot game. This is condensed in the two following propositions.

Proposition 3 When 1 = 2 , the Stackelberg-Walras equilibrium coincides with the Cournot-Walras equilibrium. Proof. We verify that the equilibrium strategies of both oligopolists, the market price and the SWE allocations reached for 1 = 2 correspond to the equilibrium strategies, the market price and the allocations that would be obtained in an environment where both producers would play a Cournot game, all the consumers owning good 2 still behaving competitively. A Cournot-Walras equilibrium for the economy is given by a market price p( e 11 ; e 21 ), a pair of strategies ( e 11 ; e 21 ) and an allocation ( x 1 ; :::; x h ; ::

:; x n+2 ) 2 R 2(n+2) + such that: (i) x h = x h (
( 1 + 2 ) ( 1 + 2 ) 2 ; 1 2 ( 1 + 2 ) ( 1 + 2 ) 2 . The CWE is ( x 11 ; x 12 ) = 1 1 2 1 + 2 2 ; (1 ) 2 1 + 2 2 , ( x 21 ; x 22 ) = 1 2 1 1 + 2 2
;

(1 )

1 1 + 2 2
and for h = 3; :::; n+2, ( 12)-( 14) lead to (s 11 ; ẽ21 ) = 4 ; 4 and p = 2 ; and ( 15)-( 17) yield (x 11 ; x12 ) = Switching from follower to leader, agent 1 does not modify anything when her/his leading position is not backed up by an objective competitive advantage (such as a better productivity) compared to agent 2.

x h1 ; x h2 ) = 1 ( 1 + 2 ) n ; 1 n . When 1 = 2 = , (
Proposition 4 When 1 < 2 , the Cournot-Walras equilibrium is not Pareto dominated by the Stackelberg-Walras equilibrium. Proof. The utility levels reached by the two Cournotian oligopolists and the competitive agents are respectively

U 1 = ( ) + 2 ln 2 ln 1 2 ln( 1 + 2 ), U 2 = ( ) + 2 ln 1 ln 2 2 ln( 1 + 2 )
and U h = ( ) ln ln n ln( 1 + 2 ), 8h = 3; :::; n + 2. For the …rst oligopolist, we have

Ũ1 U 1 = 2 ln( 1 + 2 ) ln 1 ln 2 2 ln 2 > 0 since ( 1 2 ) 2 > 0. For the other oligopolist, we have Ũ2 U 2 = 2 ln( 1 + 2 ) + 2 ln(2 1
2 ) 4 ln 1 2 ln 2 < 0. For all the other agents, this yields Ũh U h = ln( 1 + 2 ) ln 1 ln 2 > 0.

QED.

When agent 1 bene…ts from a competitive advantage ( 1 < 2 ), s/he does better as a leader than as a follower; and agent 2 is better o¤ confronted to a Cournot competitor than to a Stackelberg one. Moreover, the welfare of agents endowed with good 2 is higher when they face a Stackelberg competition and not a Cournot one.

The Stackelberg-Cournot equilibrium

In the SCE framework, it is considered that all agents behave strategically, with agent 1 as the leader. The set of followers now also encompasses all the agents endowed with good 2. The pure strategy of agent h, h = 3; :::; n + 2, is e h2 and h obtains in exchange of e h2 a quantity p2 p1 e h2 of good 1.

De…nition 5 A SCE for is given by a market price p(ŝ 11 ; ê21 ; êh2 ), a (n + 2)tuple of strategies (ŝ 11 ; ê21 ; êh2 ), with ŝ11 2 [0; ŷ11 ], ê21 2 [0; ŷ21 ] and êh2 2 [0; This equilibrium concept can be viewed as the subgame perfect equilibrium of a two-stage game. In the …rst step all the reaction functions are determined; in the second step the leader's optimal decision is made. The story is solved by backward induction, considering …rst all the strategic reactive decisions, and then the strategic active choice of the leader.

The market price which insures market-clearing is given by:

p 1 p 2 = h=n+2 X h=3 e h2 s 11 + e 21 . (21)
Taking the (n 1)e h2 , s 11 and e 21 strategies as given, each oligopolist h, h = 3; :::; n + 2, maximizes her/his utility. Following a procedure similar to the argument given in section 3, it can be shown that the strategy set can be restricted to E h , with E h = e h2 2 R + j 0 e h2 n , 8h = 3; :::; n + 2. Consequently, the utility level reached by any oligopolist h, h = 3; :::; n + 2 is given by V h = ln p2 p1 e h2 + (1 ) ln

which gives the following reaction functions:

e h2 = n , 8h = 3; :::; n + 2, (23)
where

As for ( 8)-( 9), these two reaction functions depend on s 11 and not on y 11 . We can verify that y 21 > e 21 > 0. Additionally, @e21 @s11 > 0 for s 11 2 h 0; 4 2 i and @e21 @s11 < 0 for s 11 2 h 4 2 ; 2 i . We also have @ 2 e21 @s 2 11 < 0. The same last three properties hold for y 21 (s 11 ). Finally, @e21 @ 2 < 0 and @e21 @ > 0: again, the strategic supply of the follower increases when her/his productivity goes up and depends on the structure of preferences.

Considering the best responses of all the followers, the leader maximizes her/his utility:

5 Welfare and merging about Stackelberg general equilibria

Confronting the SWE with the SCE, we can notably establish a Pareto domination (when < 1) and a merging (when tends to 1).

( ) ! 0. Moreover, @ @ = ln n 1 n + 1 n
, which satis…es @ @ < 0 (> 0) for < ( > ), with = n 1 ln n ln(n 1) . Therefore ( ) < 0, which leads to Ûh Ũh < 0, 8h = 3; :::; n + 2. QED. This proposition shows that some agents may be better o¤ as price takers and worse o¤ as price makers. More generally, this con…guration reveals the existence of a cooperation failure. The SWE Pareto dominates the SCE because all the strategic behaviors o¤set each other. The contractions of exchange and of production are general (ỹ 11 + ỹ21 > ŷ11 + ŷ21 ), but nobody succeeds in getting a better price (see Prop. 7).

Proposition 9 When

! 1, the Stackelberg-Cournot equilibrium coincides with the Stackelberg-Walras equilibrium.

Proof. Consider Eq. ( 32)-(34). We remark that (32)-( 33 The merging between the SCE and the SWE is obtained when = 1, and two di¤erent circumstances can lead this parameter to take that value. The market power of agents endowed with good 2 nears to negligible when n tends to in…nity, or when the good they own becomes intrinsically useless ( ! 1). In both cases, the Cournotian behavior tends to the Walrasian one: using one's disappeared market power comes down to not using it 9 .

It is meaningful to compare the preceding equilibria with the competitive equilibrium, when all agents behave as price takers. The market price, the production levels and the allocations for consumers h = 3; :::; n + 2 are respectively, when 1 < 2 (see Appendix 2):

p 1 p 2 = 1 . ( 38
) (y 11 ; y 21 ) = 1 ; 0 , 8h = 1; 2, (39)
(x h1 ; x h2 ) = 1 1 n ; 1 n , 8h = 3; :::; n + 2. ( 40
)
Proposition 10 Consider y as the aggregate level of production for each equilibrium. When 1 2 , y < y < y and ŷ < ỹ < y . In particular, when 1 = 2 = , ŷ = y < ỹ = y < y .

Proof. From (39) we have y =

21 = ( 1 + 2 ) 2 ( 3 1 + 3 2 ) + 1 2 2 + 2 1 2 and y = y 11 + y 21 = ( 1 + 2 ) 2 h 1 + 2 + 2 1 2 + 2 2 1 i
, then y = y. Firstly, we verify that y is the highest level of production when 1

2 . When 1 < 2 , y > ỹ if and only if 2 2 1 + 2 2 < (1 + 2 ) 1 2 , which is always veri…ed for 2 (0; 1). When 1 = 2 , y > ỹ if and only if < 1. Additionally, when 1 < 2 , y > y if and only if 1 ( 2 1 2 2 ) < (1 ) 3
2 , which is always veri…ed for 2 (0; 1). When 1 = 2 , y > y if and only if < 1. Secondly, when 1 2 , ỹ > ŷ and y > y, with ŷ = ỹ and y = y, since < 1. Thirdly, when 1 = 2 = , we have ỹ = y = (1+ ) 2 and ŷ = y = (1+ )

2

. QED. To put it in a nutshell: the more widespread the Walrasian behavior, the bigger the aggregate production. The competitive level or production is the highest one, and accordingly the competitive equilibrium price is the lowest one, as p < p = p < p = p when 1 < 2 , and p < p = p = p = p when 1 = 2 .

Conclusion

In the previous economy, trade is necessary to production and production is necessary to consumption for both oligopolists. Two types of asymmetries are here involved, in a general equilibrium framework under strategic interactions. The …rst one is an asymmetry within a sector, which captures the usual strategic interactions between an active leader and a follower. The second one is an asymmetry across sectors: agents of the …rst sector are producers, while the remaining agents of the other sector are consumers. But one salient feature is that agents endowed with no good will yet be able to exert market power through production activities. These two exchange and production asymmetries can lead to imperfectly competitive behaviors that create market distorsions.

The paper could be extended in the following directions. First, it could be interesting to increase the number of sectors in the economy, to generalize production activities in all sectors, and also to introduce non linearities in technologies. Second, optimal taxation policy could also be introduced in order to determine the conditions under which market distorsions caused by strategic interactions could be dampened.

A Appendix 1

We here follow an argument given by [START_REF] Gabszewicz | Oligopoly equilibria in exchange economies[END_REF] for Cournot-Walras equilibria in pure exchange economies, in order to show that the strategy set of both oligopolists can be restricted. We here consider the follower, the same analysis prevailing for the leader. ) , which represents the level of utility reached by the follower at the optimal solution in E 21 . Therefore, to determine the Stackelberg-Walras equilibrium, the strategy set of the follower may be restricted to E 21 . In this set, the optimal solution to problem ( ) is given by the solution to M ax

4 2 ; 2 i

 422 , where strategic substituabilities occur. Conversely, strategic complementarities are displayed for s 11 2 h 0; 4 2 i .

4 ,; 1 n,8h = 3 ;

 413 (x h1 ; xh2 ) = 1 2 n :::; n + 2. For 1 = 2 = , we …nally verify that ( e 11 ; e 21 ) = (s 11 ; ẽ21 ) and p = p, which leads to ( x h1 ; x h2 ) = (x h1 ; xh2 ), 8h. QED.

  Taking the strategy s 11 and the n strategies e h2 as given, the Cournotian follower who produces good 1 solves the program: Arg max fe21;y21g ln (y 21 e 21 ) + (1 ) ln s 11 + e 21 e 21 2 y 21 , (24) which gives the reaction functions: e 21 (s 11 ) = s 11 +

1 (x 11 ; 1 (x 21 ;

 111121 ) can be written (x 11 ; x12 ) = (x 11 ; x12 ) and (x 21 ; x22 ) = (x 21 ; x22 ). Then lim !x12 ) = (x 11 ; x12 ) and lim !x22 ) = (x 21 ; x22 ). And from (34), lim ; xh2 ), 8h = 3; :::; n + 2. QED.

1 .

 1 Consider …rst the other levels of production given in (12)-(13): ỹ = ỹ11 + ỹ21 = 2 1 2 ) 1 2 and (28)-(29): ŷ = ŷ11 + ŷ21 = ỹ. Moreover, from the proofs of Prop. (1) and (3), we know y = y 11 + y

V 2

 2 (e 21 ; y 21 ) = ln (y 21 e 21 ) + (1 ) ln p1 p2 e 21 2 y 21 .

  2 R + j 0 s 11 y 11 g and E 2 = fe 21 2 R + j 0 e 21 y 21 g 4 . And their production sets are given by Y 1 = 11 ; s 11 ) ;with y 11 2 Y 1 , s 11 2 S 1 ; and a pure strategy for oligopolist 2 is a pair (e 21 ; y 21 ), with y 21 2 Y 2 , e 21 2 E 2 . Hence, the pro…t of each oligopolist writes respectively 1 (s 11 ; y 11 ) = p 1 s 11 p 2 1 y 11 and 2 (e 21 ; y 21 ) = p 1 e 21 p 2 2 y 21 . The allocations (x h1 ; x h2 ), 8h = 1; 2, resulting of the market process follow. Agent 1 obtains in exchange of s 11 a quantity p1 p2 s 11 of good 2, and …nally consumes an amount x 12 = 1 p2 of this good and a quantity x 11 = y 11 s 11 of good 1. Similarly, agent 2 obtains in exchange of e 21 a quantity p1 p2 e 21 of good 2, and …nally consumes x 22 = 2

	Y 1 =	n (y 21 ; z 22 ) 2 R 2 + j y 21	1 2	z 22	n (y 11 ; z 12 ) 2 R 2 + j y 11	1 1	z 12	o	and by

o

. A pure strategy for oligopolist 1 is a pair (y

  p1 p2 e 21 2 y 21 (see Appendix 1). The follower maximizes V 2 (e 21 ; y 21 ) with respect to e 21 and y 21 :

	Arg max fe21;y21g	ln (y 21 e 21 ) + (1	) ln s 11 + e 21	e 21	2 y 21 .

  p1 p2 s 11 1 y 11 . This function has to be maximized with respect to s 11 and y 11 :

	Arg max fs11;ỹ11g	ln (y 11 s 11 ) + (1	) ln s 11 + e 21 (s 11 )	s 11	1 y 11 ,

  e 11 ; e 21 ; p), 8h, (ii) M axU h ( x h ) s.t. p x h p 2 ! (x 2 ( e 11 ; e 21 ; p)), 8e 21 , and (v) U 1 ( x 1 ( e 11 ; e 21 ; p)) U 1 (x 1 (e 11 ; e 21 ; p)), 8e 11 . Consider the maximization programs as given by (6) and by (10) with s 11 e 11 and e 21 (s 11 ) e 21 . The equilibrium strategies ( e 11 ; e 21 ) are solutions of the two equations e 11 = e 21 + + 2 ) 2 . Then p = 1 + 2 . The production strategies are thus ( y 11 ; y 21 ) =

		q	e21 1	and e 21 = e 11 +	q	e11 2	. This leads to
	( e 11 ; e 21 ) =	2 ( 1 + 2 ) 2 ;
		2
		1

h2 , h = 3; :::; n + 2, (iii) e 11 + e 21 = P h x h1 (p), h 6 = 1; 2, 8e 11 ; 8e 21 , (iv) U 2 ( x 2 ( e 11 ; e 21 ; p)) U 2 1 ( 1

  (ŝ 11 ; ê21 ; êi2 ; p), 8h, 8i = 3; :::; n + 2, (ii) s 11 + e 21 = 1 p P h e h2 , h 6 = 1; 2, 8s 11 , 8e 21 and 8e h2 (iii) U h (x h (ŝ 11 ; ê21 ; êh2 ; ê h2 ) U h (x h (ŝ 11 ; ê21 ; e h2 ; ê h2 )), 8e h2 , 8h = 3; :::; n + 2, (iv) U 2 (x 2 (ŝ 11 ; ê21 ; êh2 ) U 2 (x 2 (ŝ 11 ; e 21 ; êh2 )), 8e 21 , 8h = 3; :::; n + 2 and (v) U 1 (x 1 (ŝ 11 ; ê21 ; êh2 ) U 1 (x 1 (s 11 ; e 21 (s 11 ); e h2 (s 11 )), 8s 11 , 8h 6 = 1; 2.

	2(n+2) +	such that: (i) xh =
	x h

1 n ], and an allocation (x 1 ; x2 ; :::; xh ; :::; xn+2 ) 2 R

  The program of the follower, called ( ) ; consists in solving M ax (x21;x22) ln (y 21 e 21 + x 21 ) + (1 ) ln x 22 s.t. p 1 x 21 + p 2 x 22 2 (e 21 ; y 21 ) and x 21 0 and x 22 0. First, the positivity constraints on pro…ts imply that 2 (e 21 ; y 21 ) 0, which leads to p2 p1 2 y 21 e 21 . Moreover, given y 21 , any utility level that can be reached choosing e 21 y 21 can also be reached by determining e 21 in such a way that the quantity y 21 e 21 kept for later consumption is at least equal to the competitive demand of good 2, that is y 21 1 22 s.t. p 1 x 21 + p 2 x 22 p 1 y 21 2 p 2 y 21 . Accordingly, we consider only strategies (e 21 ; y 21 ) satisfying the constraint e 21 y 21 1 + 2 solution to ( ) in E 21 coincides with the solution to ( ) in E 21 , and this latter solution is given by (x 21 (p; e 21 ); x 22 (p; e 21 )) = 0; p1 p2 e 21 . If e 21 y 21 1 + 2 p2 p1 , the solution to ( ) in E 21 is thus (x 21 ; x 22 ) = e 21 (y 21 e 21 + x 21 ; x 22 ) = U 2 y 21 1 22 . Now consider that, if e 21 > y 21 1 + 2 p2 p1 , the strategy e 21 is substituted by the strategy e 0 21 = y 21 1 + 2 p2 p1 . Then e 0 21 2 E 21 and, according to the fact that (x 21 ; x 22 ) = 0; 2 p2 , the solution to ( ) in the strategy set E 21 is thus (x 21 (p; e 0 21 ); x 22 (p; e 0 (y 21 e 0 21 ; x 22 ) = U 2 y 21 1

			2	p2 p1 , in solving M ax (x21;x22)	ln x 21 +
	(1 Consider E 21 = ) ln x p2 p1 . n + : p2 p1 2 y 21 e 21 y 21 1 + 2 o p2 p1 . If e 21 y 21 e 21 2 R 2 1 + 2 p2 p1 , the h (1 ) + 2 p1 p2 i ; p1 p2 y 21 1 + 2 p2 p1 . The follower reaches the util-
	ity U 2 2	p2 p1 ; x 21 )) =
	0; p1 p2 y 21 1 + 2	p2 p1	, so U 2 2	p2 p1 ; x 22 (p; e 0 21

The Stackelberg general equilibrium concepts were proposed for the …rst time in[START_REF] Julien | Oligopoly equilibria "à la Stackelberg" in pure exchange economies[END_REF],[2008] for the revised version), and developed in the simple framework of a pure exchange economy.

The existence and the uniqueness analysis are beyond the scope of this paper. The existence of general oligopoly equilibrium usually rises speci…c problems[START_REF] Bonnisseau | Oligopoly equilibria in large, but …nite, linear exchange economies[END_REF],[START_REF] Gabszewicz | Strategic multilateral exchange. General equilibrium with imperfect competition[END_REF]).

measure the productivities.

As in[START_REF] Gabszewicz | Oligopoly equilibria in exchange economies[END_REF], this speci…cation is su¢ cient to capture some relevant features of strategic interactions in a general equilibrium framework with production.

We will denote the supply strategy of agent 1 by s 11 when s/he acts as a Stackelberg leader and by e 11 when s/he acts as a Cournot oligopolist.

We can notice that for = 0 the economy is autarkic; and for = 1 commodity 2 is a pure input and agents only consume good 1.

The Cournot-Walras equilibrium as a subgame perfect equilibrium of a two stage game was initially proposed for the by[START_REF] Busetto | Cournot-Walras equilibrium as a subgame perfect equilibrium[END_REF].

The utility Ũ2 can be written Ũ2 = ( ) 2 ln 2 + 2 ln 2 2 1 ln 2 .

The two results obtained comparing the SCE and the SWE correspond to analogous comparative results obtainable about the CE and the CWE.

which gives the vector of equilibrium strategies of the leader:

(ŝ 11 ; ŷ11 ) = 4 ;

(1 + ) 4

We deduce the followers'strategies ê21 (ŝ 11 ), ŷ21 (ŝ 11 ) and êh2 :

(1

(29) êh2 = n ; 8h = 3; :::; n + 2:

(30)

The equilibrium price is then:

The equilibrium allocations are:

,

(32)

The corresponding utility levels are:

Proposition 6 When 1 = 2 , the Stackelberg-Cournot equilibrium coincides with the Cournot equilibrium.

Proof. We verify that the equilibrium strategies of both oligopolists, the market price and the SCE allocations reached for 1 = 2 correspond to the equilibrium values that would be obtained in an environment where all agents would play a Cournot game. A Cournot equilibrium for the economy is given by a market price p( e h ), 8h, a vector of strategies ( e 11 ; e 21 ; :::; e h2 ; :::; e hn+2 ) 2 R n+2 + and an allocation ( x 1 ; :::; x h ; ::

)

for the …rst producer,

;

(1 )

for the second one, and Proof. Using (35)-(37) and the second part of the preceding proof (from which the utility levels reached at the Cournot equilibrium are deduced), we have

Proposition 4 con…rms proposition 2: the shift of agent 1 from a Cournot to a Stackelberg behavior is improving for her/him, deteriorating for her/his follower competitor (on the same side of the market) and improving for the other agents (on the other side of the market).

B Appendix 2

We here determine the competitive equilibrium for the economy . Each agent who owns good 2 solves the program M ax (x h1 ;x h1 ) ln x h1 + ( 1) ln x h2 s. p1 . The corresponding allocations are (x h1 ; x h2 ) = (0; 0), 8h = 1; 2, and (x h1 ; x h2 ) =