Philippe Jehiel

Jakub Steiner

Mark Dean

Andrew Ellis

Alessandro Pavan

Philip Reny

Larry Samuelson

Colin Stewart

Balazs Szentes

Selective Sampling with Information-Storage Constraints *

Keywords: bounded rationality, information processing, stochastic choice, confirmation bias, speed-accuracy complementarity, probability weighting, salience. JEL codes: D8, D9

A memoryless agent can acquire arbitrarily many signals. After each signal observation, she either terminates and chooses an action, or she discards her observation and draws a new signal. By conditioning the probability of termination on the information collected, she controls the correlation between the payoff state and her terminal action. We provide an optimality condition for the emerging stochastic choice. The condition highlights the benefits of selective memory applied to the extracted signals. Implications-obtained in a simple class of binary problemsinclude (i) confirmation bias, (ii) speed-accuracy complementarity, (iii) overweighting of rare events, and (iv) salience effect.

Introduction

Economic agents often acquire information about the state of the economy before making their decisions. The information is typically modelled as a signal that helps the agent refine the distribution of the state and improve the decision-making. Often, signals come over time and agents can absorb only a small number of them. We capture this information-processing friction by assuming that agents receive as many signals as they wish but can remember only a finite number of them when making their choices. In the simplest setting we analyze, the agent can only remember one signal. A key strategic variable that we consider is to allow the agent to ignore some signals with positive probability and restart the signal extraction process. We allow agents to employ an arbitrary stationary decision process that specifies for each possible signal realization a probability with which the agent restarts the process as well as the chosen action in case of termination. We do not impose time constraints and costs in the basic formulation so that the friction comes solely from the limited information-storing capacity of the agent.

We ask ourselves: Should the agent optimally make her choice as soon as she receives the first signal whatever the realization of it is, or could she be better off by rerunning the very same information-acquisition process? Can hesitation-selective repetition of a fixed stochastic decision procedure when the agent cannot aggregate information-be welfare-enhancing?

A general insight is that selective rerunning of the primitive decision procedure is typically optimal. To document this most generally, we provide a simple necessary condition satisfied by the optimal rerunning strategy. The result is an interim indifference condition imposed on the agent who has concluded her decision-making with a plan to choose a particular action. Given the recommended action, the agent's posterior expected payoff from implementing this action must be the same as the posterior expected payoff from rerunning the whole decision-making-the whole selective repetitions of the primitive signal extraction-and implementing whichever action the second run of the decision-making will recommend. We refer to this as to the second-thought-free condition.

For illustration, consider a binary decision of whether to make an investment of a fixed size.

The agent receives payoff 1 if she invests in the good state of the economy, payoff -2 if she invests in the bad state, and receives 0 when she does not invest whatever the state. Both states are a priori equally likely and give rise to a population of good and bad signals, with the share of the good signals at 90% in the good state and 10% in the bad state. The agent draws possibly several signal realizations in sequence but remembers only the last one when making her investment choice. As follows from simple optimization considerations, assume she invests if and only if the last remembered signal was good. Observe that the decision rule generated by the immediate termination upon the first signal that comes in does not generate a second-thought-free choice rule: An agent whose first observed signal was good prefers to rerun the decision process, since the new run will either lead to investing again or will give rise to the signal realization that conflicts with the first observation and will lead to not investing. Since, conditional on two conflicting signals, not investing is preferred, the agent benefits from having second thoughts when the first observed signal is good.

In the sequel, we interpret the probability of terminating the decision process after receiving a particular signal as a search intensity for the signal. A higher probability of termination at a given information set inflates the likelihood that the agent makes the terminal choice at the set.

We show that the failure of the second-thought-free condition with uniform search intensity in the above investment decision example indicates that relative to the uniform search, the agent benefits from decreasing the search intensity for the good attribute. More generally, the second-thought-free condition follows from the first-order condition imposed on the optimal search intensities.

The model provides microfoundations to a range of behavioral stylized facts. The unifying principle of our behavioral insights is the intuition that the agent targets her search towards the type of evidence that would provide her with more informed posteriors under the uniform search. This principle generates confirmation bias, since evidence that confirms the agent's prior leads to more informed posterior than does evidence that contradicts the prior. An optimally targeted information search also generates speed-accuracy complementarity; that is, accuracy of choice declines with the response time. The effect is generated by the confirmation bias: The agent encountering evidence contradicting her prior is likely to disregard the evidence and to have a second thought. Hence, long response times indicate a surprising state of the world, and the constrained-optimal choice rule commits errors in the surprising state relatively often. Overweighting of rare events occurs when the agent's task is to form a probability belief about an event that is known to be rare, such as a flight accident, by observing a random flight outcome. Since observing a flight accident is far more informative about the probability of future accidents than observing an uneventful flight, the agent optimally biases her search towards eventful flights. In the last behavioral application, we show that distinct states of the world are salient in the sense that they attract the agent's attention (i.e., trigger higher termination rates in our framework). The effect arises because an indistinct perception stimulus that can be generated by several similar states is less informative than a distinct perception stimulus that is most likely generated by a specific distinct state. Hence, the optimal information search targets stimuli indicating distinct states.

Our leading interpretation of the model is in terms of a single-person with information storage limitations but perfect ability to adjust optimally her termination strategies as a function of what she remembers. These adjustments can be viewed as a result of a successful trial and error process or as a result of evolutionary pressures in which case the adjustments may not be fine-tuned to each specific problem.1 Alternatively, one may think of the termination strategies and the final decisions as being chosen by different persons and only the one in charge of the final decision would be subject to information storage limitations, thereby ensuring that the termination strategies are optimally determined.

There is a wide range of studies that propose different modeling of optimization over information structures. Relative to rational inattention models [START_REF] Sims | Implications of rational inattention[END_REF], we provide a procedural micro-foundation of our set of the feasible information structures (which must be obtained through variations of the termination strategies and as such can straightforwardly be related to the time dimension of the decision process). Perhaps surprisingly, our memoryless agent has a similar flexibility in her choice of information structures as in the sequential-sampling model of [START_REF] Wald | Sequential tests of statistical hypotheses[END_REF] with perfect information aggregation but our approach allows for a simple derivation of the speedaccuracy complementarity which is less immediate to obtain with drift diffusion models (see however [START_REF] Fudenberg | Speed, accuracy, and the optimal timing of choices[END_REF]). Our approach is simpler than most of studies based on finite automata [START_REF] Hellmann | Learning with finite memory[END_REF][START_REF] Wilson | Bounded memory and biases in information processing[END_REF] in that it focuses solely on the optimal determination of termination probabilities (as a function of the observed signal). This allows us to derive simple necessary conditions that apply to all models in which such termination strategies are optimally determined, which turns out to be all that is needed to derive connections to psychological biases as reported above.

This paper belongs to a growing economic literature that explains behavioral stylized facts as the constrained-optimal behavior of decision-makers facing information processing frictions. For instance, [START_REF] Robson | The biological basis of economic behavior[END_REF], [START_REF] Rayo | Evolutionary efficiency and happiness[END_REF], [START_REF] Netzer | Evolution of time preferences and attitudes toward risk[END_REF][START_REF] Khaw | Risk aversion as a perceptual bias[END_REF] provide microfoundations for risk attitudes; Gabaix and Laibson (2017) endogenize discounting;[START_REF] Wilson | Bounded memory and biases in information processing[END_REF], [START_REF] Compte | Belief formation[END_REF], and Leung (2017) establish constrained-optimal ignorance of weakly informative news.2

Model

An agent faces a decision under uncertainty. She chooses an action a ∈ A and receives a payoff u(a, θ) in the fixed payoff state θ ∈ Θ drawn from an interior prior π ∈ ∆(Θ). The sets Θ and A are finite. The agent chooses a Blackwell experiment p, where p is a family of conditional signal distributions p(x | θ) that depend on θ ∈ Θ. The experiment generates a signal realization x from a finite space X. The conditional signal distributions are fully mixed: p(x | θ) > 0 for all x, θ. We allow the agent to choose among possibly several such experiments and we let P denote the set of experiments from which she chooses. We impose no restrictions on P (other than the full-support of each p).

The agent can repeat the selected experiment arbitrarily many times, but she is unable to aggregate the information across the repetitions. Each run of the experiment is a cognition that exhausts the agent's capacity dedicated to the problem being solved. Once the agent hits the constraint at the end of the experiment, she can continue only after she unclogs her capacity by amnesia.

We model this as follows. The agent can condition the repetition of the experiment on the last observed signal realization. She chooses a vector β = (β x ) x∈X ∈ B = [0, 1] |X| \ {(0, . . . , 0)} of termination probabilities β x for each signal realization x; we call β a termination strategy. The agent

O x x terminate with σ(x) terminate with σ(x ) p(x | θ) 1 -β x p(x | θ) 1 -β x β x β x
Figure 1: For each (p, β, σ), the decision process is a Markov chain evolving on the agent's states of mind, with transition probabilities that depend on the payoff state θ. The chain begins in the state of mind O and transits to states x ∈ X with probabilities p(x | θ). The process returns to O with probability 1 -β x , or terminates with choice of a = σ(x) with probability β x . runs the experiment p for the first time, receives signal realization x 1 with probability p(x 1 | θ) and terminates the reasoning with probability β x 1 . She restarts with the complementary probability 1 -β x 1 , and receives a signal realization x 2 from a new run of the process p with probability p(x 2 | θ), terminates with probability β x 2 or restarts with probability 1 -β x 2 , and continues until she terminates after a random number of repetitions of p; see Figure 1. When the agent chooses having distinct β x for different x, then she implements the familiar idea of selective memory; some facts and observations are easily forgotten whereas others are remembered and they trigger choice.

After the agent terminates the reasoning with a terminal signal realization x, she selects an action a = σ(x) according to an action strategy σ : X -→ A.3 Let S be the set of all mappings from X to A.

By excluding the termination strategy (0, . . . , 0) we force the agent to make a decision a ∈ A.

Since β = (0, . . . , 0) and each feasible experiment p generates all signal values with a positive probability in each state, the decision process almost surely eventually terminates.

The outcomes of distinct runs of p are conditionally independent. Thus, the probability that the agent terminates after t repetitions of the experiment p resulting in the signal history

x t = (x 1 , . . . , x t ) is ρ x t | θ; p, β = β xt p(x t | θ) t-1 l=0 (1 -β x l ) p (x l | θ) .
(1)

We let

r(a | θ; p, β, σ) = t x t :σ(xt)=a ρ x t | θ; p, β (2)
denote the probability that the agent who employs the experiment p, the termination strategy β, and action strategy σ terminates with action a. We call r(p, β, σ) := (r(a | θ; p, β, σ)) a∈A,θ∈Θ the choice rule. The set of feasible choice rules is R(P) = {r(p, β, σ) : p ∈ P, β ∈ B, σ ∈ S}. Sometimes we abuse notation, omit p, β, σ and write r(a | θ) for the probability of a in state θ under the rule constructed by some p, β, σ.

The repeated-cognition problem is to select, for a given prior π, utility function u and set P, a feasible choice rule r that maximizes the expected payoff: max r∈R(P) θ∈Θ,a∈A

π θ r(a | θ)u(a, θ). (3)
The optimization in (3) can be an outcome of selective pressures that favor successful decision procedures via cultural or biological evolution, or via competition of firms differing in their internal procedures. There are no costs to delaying the decision in our model but incorporating such costs would not affect our qualitative insights when these are not too big. We address agents with less severe memory constraints and with exponential time discounting preferences in Section 6.

Optimal cognition biases

We now derive a necessary optimality condition that the choice rule solving the repeated-cognition problem must satisfy. Generically, the condition requires the agent to engage in selective information processing-that is, to ignore some signals more often than others.

Second-thought-free choice rules

We start with a definition of second-thought-free choice rules. If the agent's decision process generates such a rule, then she has no incentive to rerun the process regardless of the action recommendation with which the process terminates. Our main result below states that the optimal rule is second-thought-free.

Let r be a generic stochastic choice rule that specifies conditional probabilities r(a | θ) of each action a ∈ A in each state θ ∈ Θ.

Definition 1. The choice rule r is second-thought-free with respect to the utility u and prior π if the agent prefers each action recommended by the rule to a new run of the rule r. That is, for each action a chosen with positive probability,

E α [u(a 1 , θ) | a 1 = a] ≥ E α [u(a 2 , θ) | a 1 = a], (4)
where the expectations are with respect to the random variables θ and a 2 , and α(θ, a 1 , a 2 ) = π θ r(a 1 | θ)r(a 2 | θ) is the joint distribution of the state and two actions consecutively generated by r.

The definition requires the agent who terminates with an action plan a to weakly prefer a to forgetting a and choosing whichever action a new run of the decision process will recommend.

Although the definition allows for the strict preference against having a second thought, the next lemma shows that if a choice rule is second-thought-free, then the agent is indifferent between terminating and the second thought.

Lemma 1. If r is second-thought-free, then (4) is met with equality for each action a chosen with positive probability:

E α [u(a 1 , θ) | a 1 = a] = E α [u(a 2 , θ) | a 1 = a]. (5)
We refer to (5) as the second-thought-free condition.

Proof. If (4) holds with strict inequality for some a chosen with positive probability, then

E α [u(a 1 , θ)] = E α [E α [u(a 1 , θ) | a 1 ]] > E α [E α [u(a 2 , θ) | a 1 ]] = E α [u(a 2 , θ)] ,
which contradicts that a 1 and a 2 are conditionally iid.

Optimality condition

We provide here a general necessary optimality condition imposed on the stochastic choice rule.

Proposition 1. If a choice rule solves the repeated-cognition problem (3), then it is second-thoughtfree and satisfies (5).

To understand the statement, consider the optimal choice rule r * generated by a process that consists of a random number of repetitions of a primitive cognition p. Once these repetitions of p terminate with a signal realization x and the agent is about to take an action a = σ(x), then, according to the proposition, she must be indifferent between a, and running the process associated with r * from scratch, where the new run of r * would involve new repetitions of p.

To prove Proposition 1, we introduce an effective experiment s(p, β). While the primitive experiment p(x | θ) specifies the probability that its one run results in signal x, we define s(x | θ; p, β)

to be the probability that selective repetitions of p according to the termination strategy β terminate with x. Relative to p(x | θ), the effective probabilities s(x | θ; p, β) are inflated for those x at which the agent terminates with a high probability β x .

Lemma 2. An agent who employs a primitive experiment p and a termination strategy β terminates with x in state θ with probability:

s(x | θ; p, β) = β x p(x | θ) x ∈X β x p (x | θ) . ( 6
)
Proof. Experiment s(p, β) satisfies the recursive formula

s(x | θ; p, β) = β x p(x | θ) + x ∈X (1 -β x ) p x | θ s(x | θ; p, β).
The first summand is the probability that the agent terminates with signal x after the first run of the experiment p. The second summand is the probability that the agent continues after the first run and terminates with x later. Solving for s(x | θ; p, β) gives (6).

The lemma implies that s(p, β) and hence also r(p, β, σ) are homogeneous of degree zero with respect to β. Thus, since we abstract from the delay costs, for any optimal termination strategy β * , αβ * for α ∈ (0, 1) is optimal too, and it generates the same optimal choice rule r * as β * . This multiplicity of implementation of the optimal choice rule would disappear in a natural approximation of our model with exponential discount factor approaching 1. Such approximation would select the quickest available decision process that implements the optimal feasible rule r * ; that is, it would impose that max x∈X β x = 1.

Proof of Proposition 1. Using ( 6), we rewrite the objective as follows.

max p∈P,β∈B,σ∈S θ∈Θ,x∈X

π θ β x p(x | θ) x ∈X β x p (x | θ) u(σ(x), θ). (7)
Let rule r(p * , β * , σ * ) solve the repeated-cognition problem. Consider an action a chosen with a positive probability and x such that σ * (x) = a and β * x > 0. The constraint β x ≥ 0 is not binding for this x, and the first-order condition of ( 7) with respect to β x is:

θ∈Θ π θ s (x | θ; p * , β * ) β * x u(a, θ) - θ∈Θ,x ∈X π θ s x | θ; p * , β * s (x | θ; p * , β * ) β * x u σ * (x ), θ = θ∈Θ π θ s (x | θ; p * , β * ) β * x u(a, θ) - θ∈Θ,a ∈A π θ r a | θ; p * , β * , σ * s (x | θ; p * , β * ) β * x u a , θ ≥ 0.
Multiplication by β * x , summation over all x such that σ * (x) = a, and division by θ π θ r (a | θ; p * , β * , σ * ) gives (4). Thus, the terminating agent weakly prefers termination to continuation. Lemma 1 implies the indifference (5).

Comment. Our agent can be viewed as having imperfect recall in the sense of [START_REF] Piccione | On the interpretation of decision problems with imperfect recall[END_REF]. Our approach corresponds to their ex ante approach, and the insight of Proposition 1 can be related to the observation in their absent-minded driver example that the ex ante optimal solution is also a (modified) multi-self equilibrium in which the decision problem is viewed as a team composed of multiple selves all sharing the decision-maker's objective.

Analytical solution of the binary setting

The action and state sets are binary: A = Θ = {0, 1}. To avoid a trivial case, we assume that neither action is dominant. Then, without loss of generality, u(a, θ) = u θ > 0 if a = θ and u(a, θ) = 0 otherwise. State θ is drawn from an interior prior π ∈ ∆(Θ). The set P of the feasible statistical experiments is a singleton, and the unique feasible experiment p delivers signals from X = {0, 1}.

(Lemma 5 in Appendix states that the restriction to binary signals is without loss of generality since when |X| > 2 then there exists a solution in which the agent terminates with a positive probability for at most two signals values.) We choose without loss of generality the signals labels such that the likelihood ratio p(1 | θ)/p(0 | θ) increases in θ. Signal 1 is thus relatively indicative of the state 1. We introduce a summary parameter d,

d = p(0 | 0)p(1 | 1) p(0 | 1)p(1 | 0) , (8)
and refer to d as to the perceptual distance between the two states. This summary statistics is independent of the agent's incentives and of the prior.4 Finally, a parameter R = π 1 u 1 π 0 u 0 summarizes the incentives and prior; it measures the relative a priori attractiveness of action 1.

Proposition 2.

1. When R ≥ d ( R ≤ 1/d), then the agent always chooses action 1 (action 0); 2. when R ∈ (1/d, d), then the agent chooses both actions with positive probabilities and

r * (1 | 1) = dR - √ dR (d -1)R , r * (0 | 0) = d - √ dR d -1 , (9)
β * 1 β * 0 = dR - √ dR √ dR -R p(0 | 1) p(1 | 1) . ( 10
)
Thus, for the binary setting the optimal choice rule exists and is unique. The ratio β * 1 /β * 0 of the termination probabilities is unique too, if both actions are chosen with positive probabilities. When the ex ante attractiveness of one of the actions is too strong relative to the perceptual distance of the two states, then the agent always chooses the ex ante attractive action. The decision process is non-trivial for intermediate incentives: the agent engages in repeated cognition and she chooses both actions with positive probabilities. Proof. The effective experiment s(x | θ; p, β) = s(x | θ) satisfies for any β x > 0, x = 0, 1,

s(1 | 1)s(0 | 0) s(1 | 0)s(0 | 1) = β 1 p(1|1)
x βxp(x|1)

β 0 p(0|0) x βxp(x|0) β 1 p(1|0) x βxp(x|0) β 0 p(0|1) x βxp(x|1) = p(1 | 1)p(0 | 0) p(1 | 0)p(0 | 1) = d.
Thus, s inherits the increasing likelihood property from p, and hence the identity action strategy σ I (x) = x dominates the strategy σ(x) = 1 -x. Therefore, the optimal choice rule lies in the

compact set R = {r : r(1 | 1)r(0 | 0) = dr(1 | 0)r(0 | 1)} depicted on
π 1 u 1 r * (1 | 1) = π 0 u 0 r * (0 | 0)r * (1 | 0) + π 1 u 1 r * (1 | 1)r * (1 | 1),
and two normalization conditions a r * (a | θ) = 1, for θ = 0, 1. These four conditions jointly imply ( 9) and (10) follows from ( 9) and the condition r * (1|θ) r * (0|θ) =

β * 1 p(1|θ) β * 0 p(0|θ) .

Behavioral applications

We present three behavioral effects illustrated in the binary setting from Section 4: confirmation bias, speed-accuracy complementarity, and overweighting of rare events. The fourth effect-salience of distinct states-is presented in a setting with multiple states.

Confirmation bias

Psychologists and economists distinguish at least three mechanisms leading to confirmation bias: (i) People search for evidence selectively, targeting the evidence type in accord with their priors, e.g. [START_REF] Nickerson | Confirmation bias: A ubiquitous phenomenon in many guises[END_REF]; (ii) they selectively memorize and recall the data supporting their priors, e.g. [START_REF] Oswald | Confirmation bias. Cognitive illusions: A handbook on fallacies and biases in thinking, judgement and memory 79[END_REF]; and (iii) they selectively interpret ambiguous evidence, e.g. [START_REF] Rabin | First impressions matter: A model of confirmatory bias[END_REF] and [START_REF] Fryer | Updating beliefs when evidence is open to interpretation: Implications for bias and polarization[END_REF]. We focus on the first two mechanisms and interpret them in light of our optimal repeated-cognition result.

Corollary 1. When action 1 is a priori more attractive, R ∈ (1, d), and the unique primitive experiment is symmetric, p(1 | 1) = p(0 | 0), then the agent searches relatively more intensively for signal value 1:

β * 1 > β * 0 .
Proof. Since β * 1 /β * 0 increases in R, it suffices to show that β * 1 /β * 0 = 1 when R = 1 and the primitive experiment p is symmetric. Indeed, when R = 1, then by (10),

β * 1 β * 0 = √ d p(0 | 1) p(1 | 1) = p(0 | 0)p(0 | 1) p(1 | 1)p(1 | 0) = 1,
where the last equality follows from the symmetry of p.

To see the connection to confirmation bias, consider an agent whose task is to announce the realized state of the world: she receives reward u 1 = u 0 = 1 if she makes the correct announcement and 0 otherwise. The agent finds the state θ = 1 a priori more likely than the state 0, π 1 > π 0 .

Consider the decision process that terminates immediately after the first run of the experiment and chooses the action equal to the observed signal value:

β 0 = β 1 = 1, σ = σ I .
To establish that such an unbiased process is suboptimal it suffices to show that it does not satisfy the second-thought-free condition. To see this, we examine the agent who has received the a priori unlikely signal 0 and argue that she benefits from the second thought. Such a surprised agent is better off by restarting instead of terminating with action 0, since if the new run of the process concludes with signal 0 again, then the second thought will have been inconsequential. If, however, the new run of the process concludes with signal and action 1, then the induced switch from action 0 to 1 is beneficial. This is because when the experiment p is symmetric, then, conditional on two conflicting signals, the a priori more common state 1 is relatively more likely. The agent benefits from second thought whenever she receives the surprising recommendation, and thus will deviate from the uniform search in favor of the a priori likely signal value 1.

The optimal strategy resembles the natural process in which the selective memory gives rise to confirmation bias. Consider the fastest optimal strategy, letting β * 1 = 1. When the agent observes signal 1 that confirms her prior belief, then she terminates and immediately announces the state 1. But if she is surprised, observing signal 0 that contradicts her prior, then she discards the signal with positive probability β * 0 and repeats the experiment. Although finding the exact optimal value β * 0 may be difficult, the fact that double-checking one's own reasoning when one arrives at a surprising conclusion is a common practice suggests that people are able to deviate from the unbiased information-acquisition process in the payoff improving direction.

Comments. 1) The above insight can receive an alternative political economy interpretation.

The two states θ represent left vs right wing policy. Consider a right-wing newspaper that targets the right-wing readers viewed as having a prior belief in favor of the right-wing policy. Readers can only absorb one piece of information (the analog of our information storage constraint) and the newspaper has to decide which piece of information x as generated by p(x | θ) to choose as its headline. Our model explains why such journals would target their search toward evidence favoring the right-wing policy.5

2) [START_REF] Meyer | Learning from coarse information: Biased contests and career profiles[END_REF] studies optimal biases in a sequential-learning problem of an agent who receives a sequence of signals and, unlike our agent, can aggregate the sequence. Meyer's main insight is that some asymmetries in the signal structure are optimal. Although optimal asymmetries arise both in her and our frameworks, the two papers study distinct optimizations. While our agent controls termination probabilities in a stationary decision process, Meyer's agent controls the choice of a Blackwell experiment in each round of a non-stationary process.

Speed-accuracy complementarity

Our model generates the speed-accuracy complementarity effect-a stylized fact stating that delayed choices tend to be less accurate than speedy choices; see the psychology studies of [START_REF] Swensson | The elusive tradeoff: Speed vs accuracy in visual discrimination tasks[END_REF] and [START_REF] Luce | Response times: Their role in inferring elementary mental organization[END_REF]. We establish this effect in the setting from the previous subsection:

u 0 = u 1 = 1, π 1 > π 0 , considering again a symmetric primitive experiment, p(1 | 1) = p(0 | 0) > 1/2.
Let ϕ(θ, a, t) be the joint probability distribution of the state θ, chosen action a, and the reaction time t generated by the solution (p, β * , σ I ) of the repeated-cognition problem.

Corollary 2. The probability Pr ϕ (a = θ | t) of the correct choice decreases with response time t.

Due to the stationarity of the decision process, the probability of the correct choice conditional on the payoff state is independent of the reaction time: Pr ϕ (a = θ | θ, t) = Pr ϕ (a = θ | θ). At optimum, this conditional probability of the correct choice is larger in the a priori more likely state 1 than in the state 0, reflecting the relative weights of the two states in the objective. Overall, unconditionally on the payoff state, the probability Pr ϕ (a = θ | t) of the correct choice depends on the response time because t correlates with θ. A long response time indicates that the agent has repeatedly encountered the surprising signal value 0 and has hesitated to terminate. Hence, conditional on large t, the likelihood of the surprising state becomes high. The longer the agent has hesitated, the more likely it is that she is facing the surprising state in which she is making more mistakes.

Proof. β * 1 > β * 0 by Corollary (1) since π 1 u 1 > π 0 u 0 . We let f θ = β * 1 p(1 | θ) + β * 0 p(0 | θ)
denote the probability of termination per each round in state θ. The response time t in the state θ is geometrically distributed with the decision rate andβ * 1 > β * 0 , the decision rate is higher in state 1 than in state 0: f 1 > f 0 . Thus, the likelihood ratio Pr ϕ (t | θ = 1)/ Pr ϕ (t | θ = 0) decreases with t, and hence Pr ϕ (θ = 1 | t) decreases in t. The fact that β * 1 > β * 0 , and the symmetry of p implies that the probability of the correct choice is larger in state 1 than in state 0:

f θ : Pr ϕ (t | θ) = f θ (1 -f θ ) t for t = 0, 1, . . . . Since p(1 | 1) = p(0 | 0) > p(1 | 0) = p(0 | 1)
r (1 | 1; p, β * , σ I ) = β * 1 p(1 | 1) β * 0 p(0 | 1) + β * 1 p(1 | 1) > β * 0 p(0 | 0) β * 0 p(0 | 0) + β * 1 p(1 | 0) = r (0 | 0; p, β * , σ I ) . Since Pr ϕ (a = θ | t) = Pr ϕ (θ = 1 | t)r(1 | 1; p, β * , σ I ) + Pr ϕ (θ = 0 | t)r(0 | 0; p, β * , σ I ), the result obtains.
Comment. The predictions of our model are in line with the evidence on state recognition problems reported in [START_REF] Ratcliff | The diffusion decision model: theory and data for two-choice decision tasks[END_REF] according to which 1) The posterior probability of correct recognition is higher when announcing the a priori more likely state and 2) Late announcements are relatively less precise. This is in contrast to the prediction of the traditional Wald model (see [START_REF] Fudenberg | Speed, accuracy, and the optimal timing of choices[END_REF] for an elaboration of Wald model in which the stakes attached to the correct recognition are a priori unknown).

Overweighting of rare events

We continue to consider the state-recognition task; u 0 = u 1 = 1. In contrast to previous applications, we assume that the two states θ = 0, 1 are equally likely, but the distribution of the signal values x = 0, 1 is asymmetric across states. Specifically, the probability of x = 1 in the state θ is ρ θ ∈ (0, 1) and the probability of x = 0 is 1 -ρ θ . We assume, without loss of generality, that

ρ 0 < ρ 1 < 1 -ρ 0 . 6
The a priori probability of event x = 1 is (ρ 0 + ρ 1 )/2 < 1/2, and thus the event x = 1 is relatively rarer than x = 0. The next result states that at the optimum, the agent is relatively more likely to discard the more common event x = 0 in agreement with [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF], who observe that agents tend to overweight rare events.

Corollary 3. At the optimum, the agent is biased in favor of the rare event x = 1: β * 1 > β * 0 > 0 (and her guess of the state equals the observed signal realization, i.e. σ = σ I ).

Proof. The belief formation problem studied is a special case of our binary setting with the primitive

experiment p(x | θ) = ρ θ if x = 1, p(x | θ) = 1 -ρ θ if x = 0 and
= √ d p(0 | 1) p(1 | 1) = p(0 | 1)p(0 | 0) p(1 | 1)p(1 | 0) = (1 -ρ 1 )(1 -ρ 0 ) ρ 1 ρ 0 . The inequality β * 1 > β * 0 follows from ρ 0 < ρ 1 < 1 -ρ 0 .
For illustration, consider a formation of belief over the probability of a flight accident. The accident probability per flight in the safe state of the world is 10 -6 , whereas it is 10 -5 in the dangerous state of the world, and both states are a priori equally likely. The agent can sequentially observe arbitrarily many past flight outcomes, but cannot aggregate the information, and recalls only the last observed flight. She guesses that the state of the world is dangerous if and only if the last observed flight is eventful.

Consider first an agent who always terminates right after the observation of the first datapoint. Such an agent benefits from a "second thought" whenever she observes an uneventful flight: Either the second observed flight will be uneventful, in which case the second thought will have been inconsequential, or the redrawn flight will be eventful and the agent will switch her assessment from the safe to the dangerous state. Such a switch is beneficial since conditional on two contradicting data-points the dangerous state is relatively more likely. Thus, relative to the immediate termination strategy, the agent will benefit from discarding the uneventful flight observations with positive probability.

Salience

Bordalo, Gennaioli, and Shleifer (2012) interpret salience as directed attention focus. They quote the popular work by Daniel [START_REF] Kahneman | Thinking, fast and slow[END_REF]:

"Our mind has a useful capability to focus on whatever is odd, different or unusual."

The quote states a causal relation between the two features of the salient phenomena: These are (i) odd, different or unusual, and because of (i), people benefit from (ii) focusing their attention on such phenomena. Here, we confirm Kahneman's intuition within our proposed framework. Our microfoundation of the salience effect is related to the insight emerging in psychological research on visual salience. [START_REF] Itti | Visual salience[END_REF] conceptualizes the visual salience effect as attention allocation to a subset of the visual field that is "sufficiently different from its surroundings to be worthy of [one's] attention." Similarly, in our model, a payoff state is salient if it stands out sufficiently from similar states to be worthy of the focus of the agent's information search.

As before, the agent faces a perceptual task that requires her to announce a realization of the random state θ. She is endowed with a primitive perception technology that generates a perceived value θ of the state. The primitive perception is informative but noisy: the perceived value θ equals the true state θ with a high probability, but mistakes, θ = θ, occur sometimes. We view the primitive perception technology as a black-box model of a physiological sensor that generates a noisy first impression θ of the true state θ. The agent can use the sensor repeatedly but is not able to aggregate the information. She conditions the repetition of the sensor's use on the most recent perception and announces the terminal perception.

We formalize this perception task as follows. The agent makes an announcement a ∈ A = Θ, where 2 < |Θ| < ∞, and receives payoff u(a, θ) = 1 if her announcement is correct, a = θ, and u(a, θ) = 0 if a = θ. The prior is uniform. Each use of the agent's sensor generates a signal value/perception θ ∈ X = Θ, with conditional probabilities p(θ | θ). We make two assumptions on p:

Symmetry: p(θ | θ) = p(θ | θ ). Sufficient precision: p(θ | θ) > p(θ | θ) for all θ = θ .
For two states θ 1 and θ 2 , we say that θ 1 is more distinct than θ 2 if for each state We make a simplifying assumption that the agent uses the identity action strategy σ I ; she announces the state equal to her last perception. We also make a regularity assumption that the optimal termination probabilities β x are positive for all x ∈ Θ. 7 Let r * = r(p, β * , σ I ) be the optimal feasible choice rule.

θ 3 = θ 1 , θ 2 , p(θ 1 | θ 3 ) < p(θ 2 | θ 3 ). Suppose
Proposition 3. If state θ 1 is more distinct than state θ 2 , then the agent's terminal perception is biased in favor of the more distinct state θ 1 at the expense of the less distinct state θ 2 :

r * (θ 1 | θ 2 ) > r * (θ 2 | θ 1 ).
See Appendix for the proofs omitted in the main text.

Since the primitive perception technology p is symmetric by assumption, the asymmetry in favor of the distinct state of the optimal terminal perception r * is driven solely by the optimization of the termination strategy. To gain the intuition for the salience of the distinct states, consider a state θ * that is similar to many other states and an agent who always terminates the process after the first round: β = 1. This agent is relatively uninformed whenever she forms perception θ * , since the true state differs from θ * with a sizeable probability. The agent with this indistinct perception θ * would thus benefit from "having a second thought"-i.e., from running the primitive perception formation process once again. The optimal termination strategy involves repeating the primitive process with relatively high probability whenever the agent forms a perception of an indistinct state, and this shifts the terminal perception in favor of the distinct states.

Extensions

In the first subsection, we discuss how our model can accommodate agents with more general memory constraints. Subsection 6.2 accommodates agents who discount future payoffs at an exponential rate.

Sophisticated agents

To demonstrate the flexibility of the general model, we now discuss two specific settings. They feature sophisticated agents with non-trivial memory that can be used to aggregate information over several observed signal realizations. Perhaps surprisingly, we show that those settings can in fact be interpreted as special cases of our general model that on its face value allows only for trivial memory. We show that such accommodation of non-trivial memory is possible via expansion of the set P of the primitive experiments. This allows us to establish the generality of the secondthought-free condition.

Moreover, when the state and action spaces are binary, then the setting with sophisticated agents boils down to the simple binary setting as formulated in Section 4, except for the determination of the perceptual-distance parameter d, which is now endogenously determined by the agent's ability to process information.

strategy for which some βx is zero is dominated by the strategy that immediately terminates and announces the first impression.

Example 1 (imperfect information aggregation). This setting relaxes the agent's inability to aggregate information across the repetitions of her reasoning by endowing her with a finite set of memory states that she can use to represent the signal histories. The setting of this example builds on [START_REF] Hellmann | Learning with finite memory[END_REF] and [START_REF] Wilson | Bounded memory and biases in information processing[END_REF]. The agent can repeatedly sample from a single statistical experiment that generates signal realizations from a finite signal space. Additionally, the agent is endowed with a finite set of memory states. After each run of the experiment, the agent randomizes between terminating and continuation of the decision process, where in the latter case, Let p(a | θ; γ, σ) be the probability that the agent terminates with the choice a in state θ, and let P iia be the set of all stochastic choice rules p that this agent can construct. She selects the choice rule from P iia that maximizes her ex ante expected payoff.

We now demonstrate that this example is a special case of our baseline model. Consider the baseline model with the signal space X = A and the set of the feasible primitive experiments As it turns out, no new choice rules beyond those from P iia can be constructed by these selective repetitions. This follows because the repetitions of the rule p ∈ P iia with the termination strategy β can always be replicated with an appropriate choice of a different rule in P iia that whenever p would terminate with a restarts the process from scratch with probability 1 -β a . Formally:

P = P iia . The set R(P iia ) = {r(p, β, σ) : p ∈ P iia , β ∈ B, σ ∈ S} is
Lemma 3. R(P iia ) = P iia .
According to the lemma, Example 1 is the special case of our baseline model with P = P iia and X = A, since in such a specification of the baseline model, the set of the feasible rules coincides with those in Example 1. In particular, the optimal choice rule p * ∈ P iia solving Example 1 coincides with the optimal rule r * ∈ R(P iia ) solving this specification of the baseline model.

The repeated-cognition problem with P = P iia is purely formal in that the optimal termination probabilities β * x = 1 for all x ∈ X = A, and thus the agent conducts the optimal process p * ∈ P iia only once and terminates. Nevertheless, the observation that p * solves the repeated-cognition problem has an important implication.

Corollary 4. The choice rule that solves Example 1 (imperfect information aggregation) is secondthought-free. [START_REF] Wilson | Bounded memory and biases in information processing[END_REF] differs from this example mainly in that she assumes exogenous termination probabilities. By adding optimization over the terminations to the model of Wilson, we gained the partial characterization of the optimal choice rule without fully solving the problem: One can conclude that the optimal choice rule is second-thought-free without analyzing the optimal use of the memory states.

Example 2 (partial forgetting). The agent of this example can remember up to a fixed finite number of signal realizations generated by a single statistical experiment. In each round of her decision process, she can discard a subset of the currently remembered signals values, extract a new signal realization, or terminate, where each of these decisions is determined by a stationary mixed strategy that conditions on the currently remembered stock of the signal values. The statistical experiment and the maximal number of the signals that the agent can remember determine the set of the stochastic choice rules that she can construct, from which she chooses the rule that maximizes her ex ante expected payoff. We first formalize this example as follows. Let H be the set of the signal histories h of length |h| ≤ N . The agent at a history h can (i) terminate her decision-making; (ii) discard some of the information accumulated; or (iii), if |h| < N , acquire a new signal realization. (i) An agent terminating at h chooses action σ(h). (ii) An agent who discards some information transitions to a truncation h of her current history h. 8 (iii) An agent who acquires a new signal realization transitions to a history hx, where x is the new signal realization drawn from µ(x | θ). The decisionmaking is governed by a pair of mappings γ : H × Θ -→ ∆ (H ∪ {t}) and σ : H -→ A, where γ(h | h, θ) stands for the probability that the agent at history h in state θ continues decisionmaking and transitions to h , and γ(t | h, θ) is the probability of termination at history h in state θ.

8 A truncation is obtained by deleting one or more last elements in h.

The mapping γ is constrained to satisfy

1. γ(h | h, θ) is independent of θ if h is a truncation of h, 2. γ(t | h, θ) is independent of θ, 3. γ(hx|h,θ) γ(hx |h,θ) = µ(x|θ) µ(x |θ) , 4. γ(h | h, θ) = 0
unless h is a truncation of h, or h = hx for some x ∈ X and |hx| ≤ N . Constraints 1 and 2 require the agent to condition the decision to discard information or to terminate only on her current history independently of the state. Constraint 3 allows the agent to expand her information set only by running the experiment µ(x | θ). Constraint 4 restricts each step of information acquisition to one draw from µ(x | θ) or to a partial discarding of the accumulated information. Let p(a | θ; γ, σ) be the probability that the agent who employs (γ, σ) terminates with action a in the state θ. The agent chooses γ and σ to maximize her ex ante expected payoff. As with the previous example, let R(P pf ) be the set of the feasible choice rules in our baseline model with the set of the feasible primitive experiments P identified with P pf . Lemma 4. R(P pf ) = P pf .

Thus, again, the rule p * ∈ P pf solving this example , and the optimal rule r * ∈ R(P pf ) coincide, and thus the rule solving the example must be second-thought-free.

Corollary 5. The choice rule that solves Example 2 (partial forgetting) is second-thought-free.

Additionally, when the state and action sets are binary, Proposition 2 applies to both examples with d = p * (1|1)p * (0|0) p * (0|1)p * (0|1) , and thus, relative to the simple setting with the single exogenous binary experiment, the examples have the same solution with one additional endogenous parameter d.

Impatient agents

Our baseline model abstracts from the cost of time in that the agent is only concerned with how the repetitions of the signal extraction affect the correlation of the signal with the state. We now incorporate discounting.

We continue to study the baseline model from Section 2, except that the agent discounts future payoffs exponentially with the discount factor δ ∈ (0, 1). To accommodate discounting, we redefine the choice rule induced by the experiment p, the termination strategy β and the action strategy σ as follows.

r δ (a | θ; p, β, σ) = t x t :σ(xt)=a δ t ρ x t | θ; p, β , (11)
where ρ x t | θ; p, β defined in (1) is the conditional probability of the signal history x t . That is, r δ (a | θ; p, β, σ) is the discounted probability of the choice of action a in the state θ. When δ = 1, then (11) coincides with our baseline definition of the choice rule.

The set of feasible discounted rules is R δ (P) = {r δ (p, β, σ) : p ∈ P, β ∈ B, σ ∈ S}. The discounted repeated-cognition problem is to select a feasible rule r δ that maximizes the expected payoff: max

r δ ∈R δ (P) θ∈Θ,a∈A π θ r δ (a | θ)u(a, θ), (12)
π 1 u 1 = 10×π 0 u 0 .
The primitive experiment is symmetric: p(1 | 1) = p(0 | 0) = .9. The agent terminates immediately when she observes signal value 1, β * 1 = 1. When δ > .53, then the agent is biased towards state 1: when she encounters signal value 0, then she terminates the decision-process with a probability only β * 0 (δ) < 1 (the full curve). The dotted line is β * 0 /β * 1 from the baseline model without discounting.

where discounting is incorporated in the definition of the feasible rules.

The next result generalizes the second-thought-free condition. Let r * δ = r δ (p * , β * , σ * ) the choice rule solving the discounted repeated-cognition problem (12). Proposition 4. If the termination strategy β *

x ∈ (0, 1) is interior for all x such that σ * (x) = a, then

θ∈Θ π θ u(a, θ)r * δ (a | θ) = δ θ∈Θ,a ∈A π θ u(a , θ)r * δ (a | θ)r * δ (a | θ). (13)
The condition has the same interpretation as the second-thought-free condition in the absence of discounting. The left-hand side is the payoff for following the optimal decision process r * δ summed up across all contingencies that terminate with choice of a. The right-hand side is the payoff that the agent would get across the same contingencies if she restarted the decision process r * δ instead of the termination.

For illustration, we now revisit the confirmation bias application from Section 5.1 with an impatient agent. We find that, unless discounting is too strong, the impatient agent chooses qualitatively the same strategy as the patient one, although the impatient agent speeds up her decision-making by choosing larger termination probabilities.

The setting is as follows. The agent chooses a ∈ {0, 1} and receives u(a, θ) = u θ > 0 if a = θ, and zero reward otherwise. Action 1 is a priori more attractive than action 0; π 1 u 1 > π 0 u 0 . The agent has access to a single primitive experiment p that generates signal values in X = {0, 1}. The experiment is symmetric with probabilities p(1 | 1) = p(0 | 0) = α > 1/2. Proposition 5. The agent chooses the action equal to the last observed signal realization. She terminates her decision-making immediately after she encounters signal realization 1:

β * 1 = 1. When δ ∈ 1 α+(1-α)R
, 1 , then the agent who observes x = 0 terminates with interior probability β * 0 ∈ (0, 1) that decreases in δ. When δ ∈ 0, 1 α+(1-α)R , then the agent terminates immediately:

β * 0 = β * 1 = 1.

Summary

Agents, who cannot comprehend all facts available to them, benefit from selective attention.

We show that agents can implement a targeted information search in a process that resembles the natural phenomenon of hesitation. Like a hesitant person, the agent can, conditional on the action contemplated, decide whether she implements the action or whether she will have a second thought, and run the cognition process once more. Such hesitation can be productive, despite consisting of repetitions of the same stochastic cognition process. By conditioning the probability of the repetition on the conclusion of the reasoning, the agent controls the correlation of her terminal conclusion and the payoff state. The optimal decision process arising in our model exhibits natural hesitation patterns: The agent will have second thoughts-that is, she will repeat her cognitionwhenever the expected payoff for the currently favored choice is inferior to the expected payoff for continuing decision-making. At optimum, the agent terminating the decision-making must be indifferent between terminating with the currently contemplated action, and repeating the process.

In a sense, the condition formalizes the concept of a reasonable doubt. Abstracting from many considerations such as information aggregation across the jury members, a jury deciding a trial under common law should be, if using the optimal decision procedure, indifferent between declaring a verdict and announcing a hung jury and initiate retrial.

Let us conclude by reviewing the limitations of our main result. The central assumption-

the ability of the agent to freely repeat her decision process-may fail for several reasons. One reason is that the agent may only have access to a limited data set that constrains her to a finite number of repetitions of the primitive decision process, making the optimal termination strategy non-stationary. Another complication arises if the outcomes of distinct runs of the same cognition process are not conditionally independent as assumed in our model; this may arise if some cognition errors are systematic and are likely to emerge in distinct repetitions of the cognition. We conjecture that the second-thought-free condition holds in such a case, with the agent internalizing the correlations between the cognition runs.

A Appendix

A.1 Extension of the binary setting from Section 4 to large signal spaces

The next lemma considers the setting from Section 4 with two actions and two payoff states and allow for arbitrary signal space X with |X| > 2. There exists a solution in which the agent ignores all but two signal realizations.

Lemma 5. There exists a solution in which the termination probability β x is positive for at most two signal values x ∈ X.

Proof of Lemma 5. Assume that there exists a solution with β x positive for n > 2 signals x ∈ X.

We show that then there exists a solution with n -1 positive signals. The proposition follows from the induction on n.

Let us prove the induction step. Fix the primitive experiment p employed by the agent, let β be an optimal termination strategy for the given p, and let X be the set of signals with positive β x , and write shortly s(x | θ) for the effective experiment s(x | θ; p, β) induced by p and β. Let us abuse notation by letting s(x) = θ π θ s(x | θ) stand for the unconditional effective probability of x. For

x ∈ X let q x ∈ ∆(Θ) be the posterior belief upon terminating with x: q x (θ) = π θ s(x | θ)/s(x).

Since |X | > 2 and the state space Θ is binary, there exists a signal x * ∈ X such that q x * is in the convex hull of the posteriors q x , x ∈ X \ {x * }. Let µ x be the coefficients that decompose

q x * into q x , x ∈ X \ {x * }. That is, µ ∈ ∆ (X \ {x * }) such that q x * (θ) = x∈X \{x * } µ x q x (θ) for all θ ∈ Θ.
We will construct an alternative feasible effective experiment s(x | θ) with unconditional probabilities of x denoted by s(x) and the posteriors π θ s(x | θ)/s(x) denoted by qx (θ) such that:

s(x) =    s(x) + s(x * )µ x if x ∈ X \ {x * }, 0 otherwise, (14)
and

qx (θ) = q x (θ) for all x ∈ X \ {x * }, θ ∈ Θ. ( 15
)
Since the experiment s is more informative than s (in the sense of the Blackwell comparison), there exists a solution with this alternative feasible effective experiment s, as needed for the induction step.

It remains to construct s. Note that if an effective experiment s(x | θ; p, β) = βxp(x|θ)

x β x p(x |θ) is induced by some p and β, then for any vector of probabilities βx , the experiment

s(x | θ) = βx s(x | θ; p, β) x ∈X βx s(x | θ; p, β) = βx β x p(x | θ) x ∈X βx β x p(x | θ)
is also feasible, since it is induced by p and β = ( βx β x ) x∈X .

We claim that if

βx =    c 1 + s(x * )µx s(x) if x ∈ X \ {x * }, 0 otherwise,
where c is a constant such that βx ∈ (0, 1) for all x ∈ X, then the resulting s satisfies the properties ( 14) and ( 15). Let us check:

s(x | θ) = βx s(x | θ) x ∈X \{x * } βx s(x | θ) = βx s(x | θ) c x ∈X \{x * } s(x | θ) + x ∈X \{x * } s(x * )µ x s(x ) s(x | θ) = βx s(x | θ) c x ∈X \{x * } s(x | θ) + x ∈X \{x * } s(x * )µ x π θ q x (θ) = βx s(x | θ) c x ∈X \{x * } s(x | θ) + s(x * ) π θ q x * (θ) = βx s(x | θ) c x ∈X \{x * } s(x | θ) + s(x * | θ) = βx s(x | θ) c = 1 + s(x * )µ x s(x) s(x | θ).
The property (14) holds since for all x ∈ X \ {x * }:

s(x) = 1 + s(x * )µ x s(x) s(x) = s(x) + s(x * )µ x .
To establish the property (15), check that for all x ∈ X \ {x * } and all θ ∈ Θ:

qx (θ) = π θ s(x | θ) θ ∈Θ s(x | θ ) = π θ 1 + s(x * )µx s(x) s(x | θ) θ ∈Θ 1 + s(x * )µx s(x) s(x | θ ) = π θ s(x | θ) θ ∈Θ s(x | θ ) = q x (θ).
A.2 Proof of the result from Subsection 5.4

The next result is an auxiliary lemma used in the proof of Proposition 3.

Lemma 6. Suppose that termination probabilities β x are positive for all x ∈ Θ. Then, the optimal effective choice rule r * satisfies for any pair of states θ, θ ∈ Θ:

r * (θ | θ)r * (θ | θ) = r * (θ | θ )r * (θ | θ ). (16)
Condition ( 16) is a strengthening of the second-thought-free condition (5). It requires that the agent who has terminated the decision process with perception θ, and knows that the second run of the process r * terminates with a value θ is indifferent between θ and θ . This condition is stronger than the second-thought-free condition (5), since (5) requires ( 16) to hold only on average across all θ . This strengthening holds for the special case of this application with a symmetric experiment p.

Proof of Lemma 6. The optimal effective choice rule satisfies the second-thought-free condition (5), equivalent to:

r * (θ | θ) = θ ∈Θ r * (θ | θ )r * (θ | θ ) for all θ ∈ Θ,
which after two algebraic steps gives:

r * (θ | θ) 1 -r * (θ | θ) = θ =θ r * (θ | θ )r * (θ | θ ) for all θ ∈ Θ, θ =θ r * (θ | θ)r * (θ | θ) = θ =θ r * (θ | θ )r * (θ | θ ) for all θ ∈ Θ.
The last system of equations is formally equivalent to the system of balance conditions for a Markov chain. To see this, consider an ergodic Markov chain with transition probabilities from θ to θ equal to r * (θ | θ). The balance condition for the stationary distribution µ(θ) of this chain is Recall that if a Markov chain with transition probabilities m(θ | θ) is reversible, then its stationary distribution µ(θ) satisfies detailed balance conditions

µ(θ)m(θ | θ) = µ(θ )m(θ | θ ) for all θ = θ .
Thus, it suffices to prove that the probabilities r * (θ | θ) constitute a reversible Markov chain.

Recall that a Markov chain m(θ | θ) is reversible if and only if it satisfies the Kolmogorov criterion, which requires for all sequences of states θ 1 , θ 2 , . . . , θ n , are substituted into (17), then the terms β θ and the denominators cancel out, and hence

m(θ 2 | θ 1 )m(θ 3 | θ 2 ) . . . m(θ n | θ n-1 )m(θ 1 | θ n ) m(θ n | θ 1 )m(θ n-1 | θ n ) . . . m(θ 2 | θ 3 )m(θ 1 | θ 2 ) = 1. ( 17
r(θ 2 | θ 1 ; p, βσ I )r(θ 3 | θ 2 ; p, β, σ I ) . . . r(θ 1 | θ n ; p, β, σ I ) r(θ n | θ 1 ; p, β, σ I )r(θ n-1 | θ n ; p, β, σ I ) . . . r(θ 1 | θ 2 ; p, β, σ I ) = p(θ 2 | θ 1 )p(θ 3 | θ 2 ) . . . p(θ 1 | θ n ) p(θ n | θ 1 )p(θ n-1 | θ n ) . . . p(θ 1 | θ 2 ) = 1,
as needed.

Proof of Proposition 3. Lemma 6 implies for all pairs θ, θ ∈ Θ:

and therefore,

D θ 3 θ 1 = p(θ 1 | θ 3 ) p 1/2 (θ 1 | θ 1 )p 1/2 (θ 3 | θ 3 ) < p(θ 2 | θ 3 ) p 1/2 (θ 2 | θ 2 )p 1/2 (θ 3 | θ 3 ) = D θ 3 θ 2 .
Thus,

D θ 1 θ 1 βθ 1 + D θ 2 θ 1 βθ 2 = 1 - θ 3 =θ 1 ,θ 2 D θ 3 θ 1 βθ 3 > 1 - θ 3 =θ 1 ,θ 2 D θ 3 θ 2 βθ 3 = D θ 2 θ 2 βθ 2 + D θ 1 θ 2 βθ 1 .
Using that D θθ = 1 and D θθ = D θ θ , we have

βθ 1 + D θ 2 θ 1 βθ 2 > βθ 2 + D θ 2 θ 1 βθ 1 .
The assumption of sufficient precision of p and symmetry of p imply that D θ 2 θ 1 < 1, and thus βθ 1 > βθ 2 , as needed.

A.3 Proofs of the results from Section 6

Proof of Lemma 3. All rules feasible in P iia are feasible in R(P iia ): R(P iia ) ⊃ P iia . This is immediate since when β a = 1 for all a ∈ A, then r(p, β, σ I ) = p for all p ∈ P iia .

where the first summand is the probability that the decision process terminates with x in the first round and the second summand is the discounted probability that the process terminates with x later. Solving (19) for s δ gives

s δ (x | θ; p, β) = β x p(x | θ) 1 -δ + δ x ∈X β x p(x | θ)
.

The discounted repeated-cognition problem ( 12) is thus equivalent to max p∈P,β∈B,σ∈S θ∈Θ,x∈X

π θ β x p(x | θ) 1 -δ + δ x ∈X β x p(x | θ) u(σ(x), θ). (20)
Consider x with an interior termination probability β * x ∈ (0, 1) and let a = σ * (x). The first-order condition of the problem (20) with respect to β x is: x and summation over all x such that σ * (x) = a gives (13).

Proof of Proposition 5. Any (β, σ) that leads to a selection of only one action with certainty is dominated by the decision process that terminates after the first round and chooses an action equal to the observed signal value. Thus, both β * 0 and β * 1 are positive, and the action strategy is σ * (x) = x or σ * (x) = 1 -x. Let us show that the action strategy σ * must be the identity function σ I .

Assume for contradiction that σ * (x) = 1-x. The payoff difference between the rule r δ (p, β * , σ * ) and the choice rule that always selects a = 1 must be positive, since the latter is dominated: Further, it must hold that β * 0 = 1 or β * 1 = 1. Otherwise, if both β * 0 < 1 and β * 1 < 1, then the agent can increase both β *

x by a same factor. This preserves the conditional action distribution in each state θ and increases the decision rates in both states, and thus it is a profitable deviation.

Additionally, it must be that β * 1 = 1: Using the expressions for s δ (θ | θ; p, β) = r δ (θ | θ; p, β, σ I ), the payoff for σ I and (β 0 , β 1 ) = (β, 1) is

π 0 u 0 βα 1 -δ + δ(βα + 1 -α) + π 1 u 1 α 1 -δ + δ(α + β(1 -α)) , (21)
and payoff for σ I and (β 0 , β 1 ) = (1, β) is

π 0 u 0 α 1 -δ + δ(α + β(1 -α)) + π 1 u 1 βα 1 -δ + δ(βα + 1 -α) , (22)
The assumptions that π 1 u 1 > π 0 u 0 and that α > 1/2 imply that, for any β ∈ (0, 1), ( 21) exceeds ( 22), as needed.

It therefore remains to find β * 0 ∈ (0, 1]. If the optimal value is interior, then it satisfies (13) with a = 0: π 0 u 0 r δ (0 | 0, p, β * , σ I ) = δ π 0 u 0 r 2 δ (0 | 0; p, β * , σ I ) + π 1 u 1 r δ (1 | 1; p, β * , σ I )r δ (0 | 1; p, β * , σ I ) .

After the substitution of r δ (x | θ; p, β, σ I ) = βxp(x|θ)

1-δ+δ x β x p(x |θ) , this condition simplifies into a quadratic equation for β * 0 . When δ < 1 α+(1-α)R , then this condition does not have an interior solution and the derivative of the value (21) with respect to β 0 at β 0 = 1 is positive. Thus, in this case, the unique β * 0 satisfying the first-order condition is β * 0 = 1.

Figure 2 :

 2 Figure 2: Each point in [0, 1] 2 corresponds to a choice rule. The depicted curve is a set R of the choice rules constructible from the experiment p. The objective is linear in the choice rule, and the indifference curves are downward sloping lines. The dashed line is the tangential indifference curve and the dot depicts the solution of the repeated-cognition problem.

Figure 2 .

 2 The agent's objective is linear in r and the optimal rule is the point of tangency of R and of an indifference line. When R < 1/d or R > d, then the problem has the corner solution. When R ∈ (1/d, d), then the optimal rule r * satisfies the feasibility condition r * (1|1)r * (0|0) r * (0|1)r * (1|0) = d, the second-thought-free condition (5) (applied to action a = 1):

  with equally a priori attractive actions, R = 1. Since ρ 0 < ρ 1 , the labeling of the signals satisfies the monotone likelihood property. Since R = 1 ∈ (1/d, d), Proposition 2 implies that the agent's behavior is stochastic, both β * 0 and β * 1 are positive, and the ratio of the search intensities β * 1 /β * 0 satisfies (10). Since R = 1

  for illustration that the perceptual task involves recognition of a color from a set {azure, indigo, red}. Intuitively, the red color stands out of this set, and this is captured by the above definition. Assume that the two shades of blue are similar in that the agent's first impression confuses them in 10% of cases, p(azure | indigo) = p(indigo | azure) = 0.1, but p(θ | red) = p(red | θ) = 0.01 for θ ∈ {azure, indigo}. Then, the red color is more distinct according to our definition than either of the two blue shades.

  she may transition to a new memory state. The termination decisions and the transitions among memory states follow a stationary mixed strategy that conditions on the current memory state and the last observed signal. Once the agent terminates, she maps the last memory state and the last observed signal value to a chosen action. The feasible statistical experiment and the set of the memory states specify a set of constructible choice rules, from which the agent chooses the one that maximizes her ex ante expected payoff.The formal specification of this example follows. The agent is endowed with one Blackwell experiment µ(x | θ) with a finite signal space X and, additionally, with a finite set M of the memory states m. After each run of the experiment µ, the agent either terminates or continues with decision-making. If the agent continues, then she transitions from the current memory state to a new memory state and reruns the statistical experiment µ(x | θ). That is, the agent selects a (generalization of the) termination strategy: γ : M × X -→ ∆ (M ∪ {t}), where γ(m | m, x) is the probability that the agent in the memory state m who has observed signal realization x in the last run of the experiment µ continues with the decision-making and transitions to the memory state m , and γ(t | m, x) is the probability that such an agent terminates. The terminating agent chooses action σ(m, x) that depends both on the current memory state and on the signal realization observed in the last run of µ. The agent starts the decision-making in the memory state m 0 . A pair γ, σ induces a θ-dependent Markov chain over the memory states that eventually terminates with choice σ(m, x), where m is the last memory state and x is the last signal realization observed.

  then a set of the stochastic choice rules that can be constructed as follows. The agent runs any process p ∈ P iia , and observes a signal value/action recommendation a with probability p(a | θ). She terminates with probability β a , according to the termination strategy β = (β a ) a∈A , and upon the termination chooses an action a = σ(a), where σ ∈ S is any mapping A -→ A. She reruns the process p with probability 1 -β a , observes a new action recommendation generated by p, et cetera, until she terminates after a stochastic number of repetitions of the process p.

Figure 3 :

 3 Figure3: Confirmation bias with discounting. Action 1 is a priori more attractive: π 1 u 1 = 10×π 0 u 0 . The primitive experiment is symmetric: p(1 | 1) = p(0 | 0) = .9. The agent terminates immediately when she observes signal value 1, β * 1 = 1. When δ > .53, then the agent is biased towards state 1: when she encounters signal value 0, then she terminates the decision-process with a probability only β * 0 (δ) < 1 (the full curve). The dotted line is β * 0 /β * 1 from the baseline model without discounting.

  r * (θ | θ) = θ =θ µ(θ )r * (θ | θ ) for all θ ∈ Θ, and thus r * (θ | θ) is proportional to the ergodic probability µ(θ) of the state θ for the chain with transition probabilities r * (θ | θ).

)

  The Markov chain with transition probabilities p(θ | θ) given by the primitive experiment p satisfies the Kolmogorov criterion (17) since p is symmetric by assumption. Finally, for any positive termination strategy β, the effective choice rule r(θ | θ; p, β, σ I ) satisfies the Kolmogorov criterion too. This is because r(θ | θ; p, β, σ I ) = β θ p(θ |θ) θ β θ p( θ|θ) , and when the expressions for r(θ | θ; p, β, σ I )

  It remains to show R(P iia ) ⊂ P iia . Consider p(γ, σ) ∈ P iia constructed in the setting of Example 2 by the use of the generalized termination strategy γ(m, x), and the action strategy σ(m, x). Recall that r(p(γ, σ), β, σ) is the choice rule constructed by repetitions of the rule p(γ, σ) according to the termination strategy β = (β a ) a∈A and by applying the action strategy σ : A -→ A upon the termination. We need to show that there exists γ and σ such that r(p(γ, σ), β, σ) = p(γ , σ ). This is indeed so when the termination probability γ(t | m, x) = γ(t | m, x)β σ(m,x) , the transition probability to the original memory state m 0 is γ (m 0 | m, x) = γ(m 0 | m, x) + γ(t | m, x) 1 -β σ(m,x), which is the sum of the probabilities that the original process γ transits to m 0 and that the decision process r(p(γ, σ), β, σ) restarts after termination of p(γ, σ). Additionally, for all m = m 0 , γ ( m | m, x) = γ( m | m, x). The above choice of γ implies that the process p(γ , σ ) replicates the Markov process over the memory states under r(p(γ, σ), β, σ). Finally, to replicate the choices upon terminations, we set the action strategy σ (m, x) = σ(σ(m, x)) for all (m, x).Proof of Lemma 4. Again, trivially, R(P pf ) ⊃ P pf , since r(p, (1, . . . , 1), σ I ) = p for all p ∈ P pf .Additionally, R(P pf ) ⊂ P pf . This is indeed so because for any β = (β a ) a∈A and any σ :A -→ A, r(p(γ, σ), β, σ) = p(γ , σ ) where the termination probability γ (t | h, θ) = γ(t | h, θ)β σ(h) , the transition probability to the empty signal history ∅ is set to γ (∅ | h, θ) = γ(∅ | h, θ) + γ(t | h, θ) 1 -β σ(h), and for all h = ∅, γ ( h | h, θ) = γ( h | h, θ). Finally, the action strategy is set to σ (h) = σ(σ(h)) for all histories h. Proof of Proposition 4. We extend the definition of the effective experiment to the setting with discounting. Lets δ (x | θ; p, β) = t x t :xt=x δ t ρ x t | θ; p, β ,where ρ x t | θ; p, β is the probability of the signal history x t defined in (1). Thus, s δ (x | θ; p, β) is the discounted probability that the agent's last observed signal value is x. It satisfies the recursion: s δ (x | θ; p, β) = β x p(x | θ) + δ x ∈X 1 -β x p x | θ s δ (x | θ; p, β),

  θ∈Θ π θ s δ (x | θ; p * , β * ) β * x u(a, θ) -δ θ∈Θ,x ∈X π θ s δ (x | θ; p * , β * ) s δ (x | θ; p * , β * ) β * x u(σ * (x ), θ) = θ∈Θ π θ s δ (x | θ; p * , β * ) β * x u(a, θ) -δ θ∈Θ,a ∈A π θ r * δ (a | θ; p * , β * , σ * ) s δ (x | θ; p * , β * ) β * x u(a , θ) = 0,where we have summed over all x such that σ * (x ) = a in the second line. Multiplication by β *

π

  0 u 0 r δ (0 | 0; p, β * , σ * ) + π 1 u 1 r δ (1 | 1; p, β * , σ * ) -π 1 u 1 = π 0 u 0 r δ (1 | 0; p, β * , σ I ) + π 1 u 1 r δ (0 | 1; p, β * , σ I ) -π 1 u 1 ≥ π 0 u 0 r δ (1 | 0; p, β * , σ I ) -π 1 u 1 r δ (1 | 1; p, β * , σ I ) > 0,where the first inequality follows from the fact that any discounted choice rule satisfies a r δ (a | θ; p, β, σ) ≤ 1. Similarly, the payoff difference between the rule r δ (p, β * , σ * ) and the rule that always selects a = 0 must be positive:π 0 u 0 r δ (0 | 0; p, β * , σ * ) + π 1 u 1 r δ (1 | 1; p, β * , σ * ) -π 0 u 0 = π 0 u 0 r δ (1 | 0; p, β * , σ I ) + π 1 u 1 r δ (0 | 1; p, β * , σ I ) -π 0 u 0 ≥ π 1 u 1 r δ (0 | 1; p, β * , σ I ) -π 0 u 0 r δ (0 | 0; p, β * , σ I ) > 0.The last two inequalities imply:r δ (0 | 1; p, β * , σ I ) r δ (0 | 0; p, β * , σ I ) > π 0 u 0 π 1 u 1 > r δ (1 | 1; p, β * , σ I ) r δ (1 | 0; p, β * , σ I ) .This establishes contradiction because as shown in the proof of Proposition 4, r δ (x | θ; p, β * , σ I ) =s δ (x | θ; p, β * ) = β * x p(x|θ) 1-δ+δ x β * x p(x|θ) , and thus r δ (1 | 1; p, β * , σ I )r δ (0 | 0; p, β * , σ I ) r δ (0 | 1; p, β * , σ I )r δ (1 | 0; p, β * , σ I ) = p(1 | 1)p(0 | 0) p(0 | 1)p(1 | 0) > 1.

In the latter case, while the second-thought free conditions need not be satisfied for each problem in isolation, we would still derive that some degree of selective hesitation is optimal.

Somewhat less related is a literature that explores how exogenous errors in learning lead to behavioral biases; see coarse learning in[START_REF] Jehiel | Analogy-based expectation equilibrium[END_REF] and its application to overoptimism in[START_REF] Jehiel | Investment strategy and selection bias: An equilibrium perspective on overoptimism[END_REF]. By contrast, in our approach, the agent optimizes the error distribution given the constraints.

We do not allow for mixed action strategies since the optimum can always be achieved with a pure action strategy.

In the working paper of this paper, we endow the agent with finite set of primitive experiments and show that the agent selects the one that maximizes d.

This is to be contrasted with the reputation-based explanation of[START_REF] Gentzkow | Media bias and reputation[END_REF]. See also[START_REF] Calvert | The value of biased information: A rational choice model of political advice[END_REF],[START_REF] Suen | The self-perpetuation of biased beliefs[END_REF], and[START_REF] Che | Optimal sequential decision with limited attention[END_REF] for constrained-optimal media-bias models.

We can always achieve this by relabeling the states θ and the signals values x, unless ρ0 = ρ1 or ρ0 = 1 -ρ1.

This regularity assumption must be satisfied when p(θ | θ) is sufficiently close to one for each θ, since then any

Ludmila Matysková and Jan Šedek provided excellent research assistance. Deborah Nováková and Laura Straková have helped with English. Jakub Steiner has received financial support from the Czech Science Foundation grant 16-00703S and from the ERC grant 770652. Philippe Jehiel thanks the ERC grant n • 742816 for funding.

1

When δ > 1 α+(1-α)R , then the condition has an interior solution and the derivative of the value (21) with respect to β 0 at β 0 = 1 is negative. Thus, for this range of parameters, the unique β * 0 satisfying the first-order condition is the interior value that solves the quadratic equation, solution of which decreases in δ.


The constraint functions: an attempt to evaluate the constraint rate inside structures that undergo ordered magnetic frustration P.

The constraint functions: an attempt to evaluate the constraint rate inside structures that undergo ordered magnetic frustration P. Lacorre 1


Gilles Halin

Veronika Bolshakova

Annie Guerriero

published or not. The documents may come

Collective decision-making with 4D BIM : identification of synchronous collaborative practices and natural user interactions

• propose an efficient set of BIM/4D collaboration tools;

• increase the quality of human-computer interactions with digital decision support;

• bring new scientific, professional, educational values.

2/ 4D BIM USES & PROJECT PHASES

3/ MULTI-USER SUPPORTS

There is no fully collaborative solution for a synchronous collaboration. 4D model interactions are mostly single-user targeted, thus the multi-user aspect is an important feature to be developed. An ease of access and low appropriation time with a tool is offered by natural user interfaces (NUI), we propose to implement a multi-touch collaborative table and wall within a NUI as session equipment (Fig. a,b). NUI fosters the convergence of 4D uses with project documents. The use of the table with the 4D software (Fig. 2c) reveals new values for AEC project decision-making. The thesis research objective aims for a democratization of the 4D simulations of AEC projects usage and develop an easy access to these practices for the future actors.

4/ Collaboration personas approach

BIM allows the team to use the entire set of collective knowledge, skills and competencies as parts of collective intelligence, which is the key feature for digital project management. In order to describe a project group collective activity, we propose a collaboration persona (CP) approach [6], which is derived from the original user persona method, also called Individual Persona. CP describes a new type of persona which focuses on group goals and needs, instead of personal needs, and includes collaborative work aspects (Fig. 2d). The design of a new support will consider different actor roles throughout the collaboration process, and adapt the interface and to collaboration scenarios and 4D BIM use-cases. The design is focused on: 1. providing users with a simple 4D simulations interactions 2. creating semi-automated digital feedback on collective decision-making.

VERONIKA BOLSHAKOVA, GILLES HALIN, ANNIE GUERRIERO

COLLECTIVE DESISION-MAKING WITH 4D BIM: IDENTIFICATION OF SYNCHRONOUS COLLABORATIVE PRACTICES AND NATURAL USER INTERACTIONS

  4D (3D+time) BIM major benefits: information accessibility; clear visualization -all actors access 3D model & attached schedule and analysis. Guerriero et al. [2] propose 4D BIM uses. Their introduction and implementation by project phases and with a corresponding LOD is summarized in Fig2. 4D BIM main value is perceived as a visualization, however other potential uses have not yet been put in a common practice. All of them involve stakeholders' expertise and collaboration.

Figure 2 -Figure 1 -

 21 Figure 2 -Project phases and model LOD with 4D BIM use potential introduction [4]

Figure 2 -

 2 Figure 2 -Experiment session with AEC professionals: a.Desicion-making with 4D on a touch table Meetiiim & Shariiing [5] collaboration and natural interactions environment; b. Project documents visualisation and 2D «sketch» annotations of 4D model; c. 4D model of concrete structure construction logics; d. Syncronisation for decision making session of collaboration persona and project documents

This thesis is a part of ANR 4DCollab international research project [1]. Fig. 1 resumes related to the project research steps. The project ames to:


Rayan Charafeddine

Damien Tedoldi
email: damien.tedoldi@enpc.fr.

Philippe Branchu

Eric Thomas

Vincent Laurent

Marie-Christine Gromaire

Keywords: Metals, Road runoff, Soil contamination, Stormwater management, Traffic, Vegetative filter strips

Cette étude s'intéresse à la contamination métallique de terres d'accotements routiers (i) en caractérisant la variabilité des concentrations avec la distance de la route et la profondeur, et (ii) en recherchant des variables explicatives des différences entre les sites. Pour ce faire, 40 tronçons de routes ont été sélectionnés et échantillonnés à 5 distances de la route et deux profondeurs. Les teneurs en métaux traces ont été analysées par spectrométrie de fluorescence X. La contamination en cuivre et en zinc se concentre essentiellement en bordure de chaussée, avec des teneurs de 45 à 330 mg/kg pour le cuivre, et 150 à 1000 mg/kg pour Zn. Le plomb présente une distribution plus uniforme sur tout l'accotement avec des niveaux de contamination globalement modérés, conséquence probable de la diminution des sources de Pb avec l'interdiction de l'essence plombée. Pour chaque distance jusqu'à 1,2 m, une corrélation significative est visible entre les concentrations en Zn (ou Cu) et la densité du trafic sur la route adjacente. Pour un trafic donné, le taux de poids lourds tend à accroître la contamination de l'accotement. Ces résultats prometteurs permettent d'envisager une extrapolation à des sites non échantillonnés, afin de mieux appréhender les besoins d'entretien de ces infrastructures et potentiellement d'améliorer les pratiques de maintenance courantes.

Intra-and inter-site variability of soil contamination in vegetated roadside embankments -Implications for maintenance operations

Variabilité intra-et inter-sites de la contamination du sol sur les accotements routiers végétalisés -Implications sur les opérations d'entretien

INTRODUCTION

In road environments, stormwater management is generally achieved through a system of filter strips, ditches, detention/infiltration basins, and/or other structural Best Management Practices. It has been widely documented that the infiltration of contaminated runoff into the soil of roadside embankments results in a long-term accumulation of ubiquitous and persistent traffic-related compounds such as trace metals or polycyclic aromatic hydrocarbons [START_REF] Napier | Evidence of traffic-related pollutant control in soil-based Sustainable Urban Drainage Systems (SUDS)[END_REF][START_REF] Werkenthin | Metals in European roadside soils and soil solution -A review[END_REF]. Besides, due to the progressive formation of a humus and sediment layer at the soil surface, the infrastructure has to undergo regular maintenance -mostly consisting of surface soil scraping -so as to preserve its hydraulic functions by ensuring diffuse inflow of road runoff. Hence, the pervasive contamination of soil brings about several questions regarding the specific requirements for the disposal, reuse or treatment of the excavated materials.

Despite a relatively abundant literature about metal contents in roadside soils [START_REF] Werkenthin | Metals in European roadside soils and soil solution -A review[END_REF], it is still difficult to identify pragmatic responses to these concerns. Intra-site variability is rarely characterized at resolutions fine enough to be of operational interest [START_REF] Tedoldi | Spatial distribution of heavy metals in the surface soil of source-control stormwater infiltration devices -Inter-site comparison[END_REF]. As to intersite variability, several factors have been investigated (among which traffic density and composition, road age, vehicular speeds, and vegetative cover) but there is no consensus over their effects and their relative importance. In other words, soil contamination does not appear to be easily predictable. This may be due to the fact that the authors did not account for the spatial variability of concentrations within the system itself, thus comparing soil samples with different levels of representativeness. By using a well-designed methodology, [START_REF] Horstmeyer | Evaluation of site-specific factors influencing heavy metal contents in the topsoil of vegetated infiltration swales[END_REF] could evidence a trend between Cu, Pb or Zn contents and the annual average daily traffic, in the case of vegetated infiltration swales.

The first objective of the present work is therefore to extend this approach to the case of roadside embankments, using a proper assessment of the site-specific distribution of metals in the surface soil to generate comparable data between sites. In doing so, a second goal is pursued, which consists of identifying accurate descriptors of soil contamination among traffic characteristics and possibly pedological parameters. For that purpose, the French department of Seine-et-Marne, in the Paris region, has been retained as a study area, where an improved sampling methodology has been applied to 40 embankments alongside departmental roads with different traffic characteristics.

MATERIAL AND METHODS

Study sites

Seine-et-Marne is a semi-rural district in the Paris Region with an area of ~6000 km 2 . It contains around 4000 kilometers of roadways managed by the county council. The 40 study sites were spread across all the department's area to encompass most of its soil types and environments. They were chosen to guarantee a uniform bivariate distribution of two road factors which were deemed relevant: the annual average daily traffic (AADT) and the percentage of semi-trucks. So as to target the traffic-related sources of contamination, none of the study sites had a metallic guardrail -except two of them which were retained for comparison purposes -nor had them road signs or street lights along the sampled section. The date of the latest maintenance operation was not always known with precision, but it was in any case prior to five years for each embankment.

Sampling methodology

The sampling methodology was designed to capture the transversal trend of soil concentrations, i.e. their distribution with respect to the distance from the road: soil samples were collected at four to five distances depending on the width of each embankment (0, 0.3, 0.7, 1.2, and 1.8-3 m). Mean samples were composited from 6 different locations on a transect parallel to the road and quartered to mitigate the effects of a potential variability at any given distance (Fig. 1). The sampling depths corresponded to the upper 2 cm, and the underlying layer (2-5 cm) whenever the road sub-base was deep enough.

Sample pre-treatment and analysis

All samples were oven-dried at 40°C, ground with a pestle, then passed through a 2-mm nylon sieve. Elemental analysis was achieved via X-ray fluorescence spectrometry (Thermo Scientific, portable Niton™ analyzer XL3t). 4 to 6 homogenized subsamples were analyzed independently, thus enabling the repeatability to be checked, then each sample was assigned its average metal contents. Among the analytical range of the apparatus, copper, lead, and zinc, as well as nickel and chromium when quantifiable, were retained as tracers of traffic-derived contamination, because of their well-documented relevance in urban and highway stormwater runoff [START_REF] Huber | Critical review of heavy metal pollution of traffic area runoff: Occurrence, influencing factors, and partitioning[END_REF].

RESULTS AND DISCUSSION

Cu, Pb, Zn, and to a lesser extent Cr were consistently above the analyser's limits of quantification and thus were considered for further analysis. Cu and Zn appeared to be strongly correlated (Spearman ρ = 0.85). Their typical distribution within each site consisted of a continuous decrease in concentrations with increasing distance from the road, and a slight decrease between the surface sample and the underlying one (Fig. 2), which is consistent with previous assessments [START_REF] Werkenthin | Metals in European roadside soils and soil solution -A review[END_REF]. The median ratio between the concentrations found at 0 and 1.2 m was 2:1. These patterns were however not systematically replicated by Pb or Cr (Fig. 3); the relative uniformity of their distribution across the entire width of the embankments, as well as their moderate contents in most sites, might be a consequence of the diminishing sources of these two metals in road environments. As discussed elsewhere, the spatial variability of copper and zinc reflects the time-integrated signature of cumulative water fluxes and sediment deposition at the soil surface: it especially demonstrates that infiltration is limited to a narrow area for most rainfall events, given the prevalence of long-duration, low-intensity events in the Paris region [START_REF] Tedoldi | Spatial distribution of heavy metals in the surface soil of source-control stormwater infiltration devices -Inter-site comparison[END_REF]. Data from each site was then synthesized into different indicators to be used as a basis for inter-site comparisons. For instance, considering the significant build-up of zinc or copper near the road edge, the mean content in the first 30 cm of each embankment was assumed to constitute a relevant information about the sites' contamination levels. This indicator was plotted against AADT in the case of Zn, with colour gradients representing the percentage of semi-trucks (Fig. 4). An increasing trend was observable over the entire range of traffic, reflecting the effect of increasing emission sources -which in road environments have been shown to be essentially brake and tire wear for Cu and Zn, respectively [START_REF] Huber | Critical review of heavy metal pollution of traffic area runoff: Occurrence, influencing factors, and partitioning[END_REF] -on pollutant build-up in roadside soil. Similar trends could also be observed at every distance from the road up to 1.2 m: the slope of the regression line decreased with increasing distance. The vertical dispersion of the data points could be attributed partly to truck traffic, and probably to differences in soil characteristics resulting in variable retention properties.

Fig. 4 -Relationship between the mean content of Zn in the first 30 cm of each embankment and the traffic characteristics (AADT on the horizontal axis and %Trucks represented as a colour gradient).

Interestingly, most studies which focused on road runoff did not evidence strong correlations with traffic indicators owing to the significant inter-event variability of metals concentrations [START_REF] Huber | Critical review of heavy metal pollution of traffic area runoff: Occurrence, influencing factors, and partitioning[END_REF]. Although stochastic processes seem to govern concentrations in runoff, long-term accumulation in soil acts like an averaging process which mitigates the inter-event variability. This illustrates the interest of targeting the soil medium for such assessments, given its propensity to function as a "passive sampler" of traffic-derived metal fluxes from the road to the embankment.

CONCLUSIONS

In the present study, soil contamination in roadside embankments was investigated through an extensive sampling campaign in the Paris area. The results obtained demonstrate: (i) a good reproducibility of the "typical" intra-site distribution of metal contents; and (ii) a clear tendency between Zn and Cu levels in the most contaminated zone of each site and the annual average daily traffic of the nearby road. These results could be achieved thanks to a simplified albeit carefully designed sampling protocol which enables a proper comparison of data with identical representativeness. Further research will focus on the soil characteristics (especially organic matter content) as a possible additional source of variability.

Fig. 1 -

 1 Fig. 1 -Schematic representation of the methodology for soil sampling.

Fig. 2 -

 2 Fig. 2 -Typical distribution of Zn in 3 embankments with contrasting traffic densities.

Fig. 3 -

 3 Fig. 3 -Surface Pb contents across all sites, as a function of the distance from the road.