Nobuyuki Hanaki
email: nobuyuki.hanaki@unice.fr

Nicolas Jacquemet
email: nicolas.jacquemet@univ-paris1.fr

Stéphane Luchini
email: stephane.luchini@univ-amu.fr

Adam Zylbersztejn
email: zylber-sztejn@gate.cnrs.fr.

Juergen Bracht

Colin Camerer

Guillaume Fréchette

Haoran He

Asen Ivanov

Frédéric Koessler

Rosemarie Nagel

Ariel Rubinstein

Jason F Shogren

Jean-Marc Tallon

Antoine Terracol

Marie Claire Villeval

Keywords: Dominance solvability, cognitive skills, CRT, Raven's test, experiment

Dominance solvability is one of the most straightforward solution concepts in game theory. It is based on two principles: dominance (according to which players always use their dominant strategy) and iterated dominance (according to which players always act as if others apply the principle of dominance). However, existing experimental evidence questions the empirical accuracy of dominance solvability. In this study, we study the relationships between the key facets of dominance solvability and two cognitive skills, cognitive reection and uid intelligence. We provide evidence that the behaviors in accordance with dominance and onestep iterated dominance are both predicted by one's uid intelligence rather than cognitive reection. Individual cognitive skills, however, only explain a small fraction of the observed failure of dominance solvability. The accuracy of theoretical predictions on strategic decision making thus not only depends on individual cognitive characteristics, but also, perhaps more importantly, on the decision making environment itself.

Introduction

Consider a game in which every decision maker is faced with a nite set of choices such that one specic choice always brings him higher monetary payo than other choices, irrespective of the choices made by other players. In this situation, the individual choice boils down to going for either a higher or a lower monetary payo. The straightforward response of a decision maker who cares about his monetary payo is to disregard dominated actions i.e. actions that may only deteriorate payo relative to other actions. This dominance principle is the most basic solution concept of game theory [START_REF] Camerer | Behavioral Game Theory: Experiments in Strategic Interaction[END_REF]. It becomes very powerful when embedded in a strategic reasoning as a stepwise process. In each step, the dominance principle implies that dominated strategies should be eliminated from an agent's strategy space. In an important class of games known as dominance-solvable games this iterated elimination of dominated strategies leads to a unique solution.

Strikingly, the data collected from numerous experiments on dominance-solvable games raise important questions about the empirical accuracy of predictions derived from this principle. Subjects tend to display less strategic sophistication than is needed to justify many applications of iterated dominance (and related renements) to model human decision making in strategic environments [START_REF] Crawford | Theory and experiment in the analysis of strategic interaction[END_REF]. The beauty contest game is one of the textbook examples of this issue. 1 A given set of players is asked to choose a number in the range [0,100]. To win the game, a player should choose a number that is the closest to p = 2/3 of the average of all chosen numbers. Any number above 2/3 × 100 ≈ 66.7 violates rst-order dominance, because the average has to be lower than 100. Knowing this, players should all choose numbers no greater than 66.7, meaning that their average may not exceed 2/3 × 66.7 ≈ 44.5. This reasoning lowers the target as the number of iterations increases, eventually leading to the unique Nash equilibrium in which all players choose 0. In many experimental studies of this game, the numbers chosen by players are used as a proxy of the depth of iterated reasoning. 2 A well replicated stylized fact is to observe 1/3 of subjects choosing a number higher than 67, and at least 1/3 a number between 44 and 67.

This paper focuses on one of the earliest and simplest example of such an empirical inaccuracy of dominance solvability, adapted from a 2 × 2 game discussed in [START_REF] Rosenthal | Games of perfect information, predatory pricing and the chain-store paradox[END_REF] and rst brought to the laboratory by [START_REF] Beard | Do People Rely on the Self-Interested Maximization of Others? An Experimental Test[END_REF].

3

The normal-form representation of this game is given in Table 1. With L < S < H, m < h, and s < h, the game is one-step dominance solvable: the elimination of player B's weakly dominated strategy l immediately leads to the Pareto-Nash equilibrium (R, r). 4

In line with observed behavior in other dominance solvable games, numerous studies (summarized in Table 2) nd frequent failures to achieve the Pareto-Nash equilibrium. In spite of variations in the design (described in the table), deviations from the standard theoretical predictions are systematic and sizable. First, dominance is frequently violated by player Bs. Depending on the exact experimental setup, up to 27% column players choose a strictly dominated action.

Second, player As violate iterated dominance, even in those cases in which player Bs commonly obey dominance. As an example, while only 6% of player Bs violate dominance in Jacquemet and Zylbersztejn (2014)-ET2 and BT2, 26% of row players still contradict the predictions of dominance solvability by choosing L (and this gure may even attain 86% in other instances, see Beard,Beil Tr. 5 in Table 2). As shown in the three middle columns of the table, both the absolute and the relative size of the stakes vary a great deal from one study to the other. Several lessons emerge from this accumulated evidence. First, both players react to their own monetary incentives. Second, in some cases player As also adjust their behavior to player Bs' incentives. Finally, as shown by [START_REF] Jacquemet | What drives failure to maximize payos in the lab? A test of the inequality aversion hypothesis[END_REF], players' inecient behavior does not fade away with repetition and cannot be explained by inequality aversion (as framed by [START_REF] Fehr | A Theory Of Fairness, Competition, And Cooperation[END_REF].

The aim of the present paper is to explore whether this empirical puzzle is related to players' cognitive skills. In this sense, our investigation belongs to a recent and growing body of experimental studies in both psychology and economics which investigate the relationship between strategic 3 Both [START_REF] Camerer | Behavioral Game Theory: Experiments in Strategic Interaction[END_REF] and [START_REF] Crawford | Theory and experiment in the analysis of strategic interaction[END_REF] consider this game as a basic example of a dominance-solvable game, and a glaring case of a mismatch between theoretical predictions and actual behavior.

4 If the game is played sequentially (so that player A moves rst), the same solution can be obtained through backward induction. Note that if s > h, the solution does not change (since l remains player B's weakly dominated strategy), but the outcomes are no longer Pareto-rankable. Beard andBeil (1994), Schotter, Weigelt, and[START_REF] Schotter | A Laboratory Investigation of Multiperson Rationality and Presentation Eects[END_REF] and [START_REF] Goeree | Ten Little Treasures of Game Theory and Ten Intuitive Contradictions[END_REF] nd that this environment also generates important violations of standard theoretical predictions. Note. For each implementation in row, the rst column describes the actual design of the experiment: simultaneousmove strategic-form game (Str), simultaneous-move extensive-form game (Ext), sequential-move game (Seq). The monetary payosof each outcome, displayed in columns 2-4, are in USD in [START_REF] Beard | Do People Rely on the Self-Interested Maximization of Others? An Experimental Test[END_REF] and [START_REF] Cooper | Evidence on the equivalence of the strategic and extensive form representation of games[END_REF], in cents of USD in [START_REF] Goeree | Ten Little Treasures of Game Theory and Ten Intuitive Contradictions[END_REF], in Yens in [START_REF] Beard | Reliant behavior in the United States and Japan[END_REF], and in Euros in [START_REF] Jacquemet | What drives failure to maximize payos in the lab? A test of the inequality aversion hypothesis[END_REF]. The game is repeated ten times in changing pairs in [START_REF] Jacquemet | What drives failure to maximize payos in the lab? A test of the inequality aversion hypothesis[END_REF], and one-shot in all other instances.

behavior and cognitive skills.

the p-beauty contest game by [START_REF] Coricelli | Neural correlates of depth of strategic reasoning in medial prefrontal cortex[END_REF] suggest that strategic thinking about other players' thoughts and behavior is implemented by medial prefrontal cortex (mPFC) one of the brain areas commonly associated with theory of mind. 7 An economic experiment by Car- penter, Graham, and Wolf ( 2013) also shows that people with higher cognitive ability make more accurate predictions of others' choices in a 20-player beauty contest game. Second, people with higher cognitive skills apply more sophisticated reasoning and are more apt in strategic adaptation. [START_REF] Burks | Cognitive skills aect economic preferences, strategic behavior, and job attachment[END_REF] report that subjects with higher cognitive skills more accurately predict others' behavior in a sequential prisoners' dilemma game, and adapt their own behavior more strongly. In the context of the p-beauty contest game, subjects with higher cognitive skills are not only found to carry out more steps of reasoning on the equilibrium path [START_REF] Burnham | Higher cognitive ability is associated with lower entries in a p-beauty contest[END_REF][START_REF] Brañas-Garza | Cognitive eort in the Beauty Contest Game[END_REF], but also to adapt their behavior to their opponents' cognitive skills [START_REF] Gill | Cognitive ability, character skills, and learning to play equilibrium: A level-k analysis[END_REF] as well as to their beliefs about their opponents' cognitive skills [START_REF] Fehr | Who knows it is a game? On strategic awareness and cognitive ability[END_REF]. Third, cognitive skills may be associated with the economic eciency of outcomes of both individual and group activities. [START_REF] Corgnet | Cognitive Reection and the Diligent Worker: An Experimental Study of Millennials[END_REF] nd that higher cognitive skills predict better performance and less shirking in an experimental labor task (summing up tables of 36 numbers without using a pen). [START_REF] Jones | Are smarter groups more cooperative? Evidence from prisoner's dilemma experiments, 1959-2003[END_REF], Al-Ubaydli, [START_REF] Al-Ubaydli | Patience, cognitive skill, and coordination in the repeated stag hunt[END_REF] and Proto, Rustichini, and Soanos (2014) report that groups with higher cognitive skills attain higher cooperation rates in repeated prisoner's dilemma games. On the other hand, Al-Ubaydli, [START_REF] Al-Ubaydli | Patience, cognitive skill, and coordination in the repeated stag hunt[END_REF] do not nd a relationship between group members' average cognitive skills and the eciency of outcomes in a stag hunt coordination. 8

Our contribution is twofold. First, we provide new evidence on the relationship between strategic behavior and cognitive skills. We show that systematic mismatches between theoretical predictions and actual behavior in a classic 2×2 7 See [START_REF] Hampton | Neural correlates of mentalizingrelated computations during strategic interactions in humans[END_REF] for related evidence. [START_REF] Hanaki | Cognitive ability and the eect of strategic uncertainty[END_REF]. The main focus of that study is player As' behavior under strategic uncertainty and its relation to monetary incentives and uid intelligence. Certain elements of their design (such as the use of Human and Robot conditions and interest in players' cognitive skills) inevitably needed to be adopted in the present study in order to address a much more general question of the empirical validity of the solution concept of dominance solvability. More precisely, we are interested in both players' behavior (so as to measure the use of dominance by player Bs and the use of iterated dominance by player As under dierent information structures).

We also make a methodological contribution, since in this paper we associate players' behavior with multiple facets of cognitive skills: uid intelligence (measured by Raven's test) and cognitive reection (measured by CRT).

Our rst treatment variable is the size of the stakes, as represented by Game 1 and Game 2 in Table 3. Although they have the same strategic properties, these two game matrices dier in terms of the saliency of monetary incentives to use (iterated) dominance. In Game 2, player As may earn a surplus of only 0.25 when moving from L to (R, r) (with payo going from 9.75 to 10), while ending up in (R, l) is relatively costly (yielding only 3). In Game 1, the potential gains and losses from action R relative to L are more balanced: the gain from moving from L to (R, r) increases to 1.5 (with payo moving from 8.5 to 10), while the outcome (R, l) becomes less costly (now yielding 6.5). The incentives of player Bs, in turn, go in the opposite direction: the gain from using the dominant strategy r (and conditional on player As' choice R) is lower in Game 1 (with payo increasing from 4.75 to 5 between (R, l) and (R, r)) than in Game 2 (where payo increases from 8.5 to 10). In line with [START_REF] Jacquemet | What drives failure to maximize payos in the lab? A test of the inequality aversion hypothesis[END_REF] and Hanaki, Jacquemet, Luchini, and Zylbersztejn (2016) (who report that both players only react to their own monetary incentives) and as discussed in Section 3.1, each of these games generates sizable yet diverse empirical violations of dominance solvability. These two games together thus provide a wide range of monetary incentives to use dominance solvability within a common strategic environment. 9

Our second treatment variable is related to the nature of player B (the column player) who may be represented either by a human subject (Human condition) or a pre-programmed computer (Robot condition). The Human condition enables us to capture two cardinal breaches of dominance solvability: the failure to use the dominant strategy (player Bs' behavior) and the failure to best respond to others' dominant actions (player As' behavior). However, the latter behavior occurs under strategic uncertainty and thus might stem from two distinct sources: bounded rationality and rational behavior under uncertainty. More precisely, player As may simply have a limited capability of best responding to dominant strategy, but may also intentionally refrain from best responding when in doubt about player Bs' use of dominant strategy. To separate these two eects, we introduce the Robot condition in which a human subject acting as player A interacts with a computerized player B who is pre-programmed to always choose r. We clearly inform the subjects in the Robot condition that they are interacting with a pre-programmed computer: the computer chooses r at each round, without exception (bold in the original instruction sheet). This is the only dierence in the rules and procedures between Human and Robot conditions. 10 Thus, the key property of the Robot condition as compared to the Human condition is neutralizing strategic uncertainty player As face, while maintaining space for boundedly rational behavior.

The design of the experiment is otherwise the same in all four experimental conditions. We explore whether behavior is sensitive to learning by considering ten uniform, one-shot interactions.

In order to homogenize incentives across rounds, the following rules are implemented: all games are played in strict anonymity, roles are xed, and subjects' payos are computed based one randomly drawn round. In the Human condition, players are matched into pairs using a perfect stranger, round-robin scheme, which guarantees that subjects are involved in a series of one-shot interactions despite the repetition of the game.

11

Our control variables also include two measures of cognitive skills. Both of them are introduced as part of a post-experimental supplementary task. Subjects' participation is rewarded with extra 5 Euros; otherwise, their answers are not incentivized. 12 The supplementary task starts with a 9 Herein, we restrict our design to these two game matrices and do not seek to further investigate the eects of monetary incentives on both players' behavior. These eects are analyzed in detail in [START_REF] Jacquemet | What drives failure to maximize payos in the lab? A test of the inequality aversion hypothesis[END_REF] and [START_REF] Hanaki | Cognitive ability and the eect of strategic uncertainty[END_REF].

10 An English translation of the original instructions in French is provided as supplementary material in Section ??. 11 See [START_REF] Jacquemet | Learning, words and actions: experimental evidence on coordination-improving information, The B[END_REF] for a detailed motivation and description of this design.

12 Absence of monetary incentives for providing corrects answers is a standard procedure for both CRT and Raven's tests. Recent evidence on both tests suggests that monetary incentives do not per se aect people's performance.

See Brañas [START_REF] Garza | Cognitive Reection Test: Whom, how, when[END_REF] for a metastudy on the determinants of CRT scores and [START_REF] Eckartz | How do incentives aect creativity?[END_REF] and [START_REF] Dessi | Strong intrinsic motivation[END_REF] for experimental evidence on the role of monetary debrieng question, where subjects are asked to report any information they nd relevant about how their decisions has been made. Then, we implement the following measures of cognitive skills.

The rst task is the standard Cognitive Reection Test based on [START_REF] Frederick | Cognitive Reection and Decision Making[END_REF] which measures cognitive reectiveness or impulsiveness, respondents' automatic response versus more elaborate and deliberative thought (Brañas-Garza, García-Muñoz, and Hernán González, 2012, p.255).

It contains three questions:

1. A notebook and a pencil cost 1.10 Euros in total. The notebook costs 1 Euro more than the pencil. How much does the pencil cost?

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

Subjects are informed that this set of three questions should be answered within 30 seconds (although we allow them to provide answers even after this time has elapsed). In this way, subjects can be classied according to their overall score (that is, the total number of correct answers) which can range from 0 to 3.

The second task is Raven's progressive matrix test (often called Raven's test), a picture based, non-verbal measure of uid intelligence, that is the capacity to think logically, analyze and solve novel problems, independent of background knowledge (Mullainathan and Shar, 2013, p.48). It is widely used by, e.g., psychologists, educators and the military [START_REF] Raven | The Raven's Progressive Matrices: Change and Stability over Culture and Time[END_REF]. It consists of a series of tasks to be solved within a xed amount of time. In each task, a subject should pick a single element (among 8 options) that best ts a set of 8 pictures. The level of diculty increases from one question to the other. 13 In our experiment, each participant is given a series of 16 tasks to be solved within 10 minutes. Individual scores in Raven' test are computed as the number of correct answers to the 16 items of the test.

Experimental procedures

For each game matrix, we run three Human sessions (involving 20 subjects per session: 10 player As interacting with 10 player Bs), and two Robot sessions (involving 20 player As per session interacting with automated player Bs). Subjects are given a xed fee equal to 5 euros to compensate participation to the experiment.

incentives in Raven's test.

13 See Raven (2008) for an overview.

Upon arrival to the laboratory, participants are randomly assigned to their computers and asked to ll in a short administrative questionnaire containing basic questions about their age, gender, education, etc. Experimental instructions are then read aloud: subjects are informed that they will play multiple rounds of the same game, each round with a dierent partner, and that their own role will remain unchanged throughout the experiment. Finally, subjects are asked to answer a short comprehension quiz. Once the quiz and any questions from participants are answered, the experiment begins. After each of the ten rounds of the game, subjects are only informed of their own payos. Information about past choices and payos is updated after each round and displayed at the bottom of the screen. Take-home earnings correspond to the outcome of a single round that is randomly drawn at the end of each experimental session.

In addition, the experimental game is followed by supplementary tasks. An additional 5 euros fee is paid to each subject for completing this part. Immediately after the end of the experimental game, participants are provided with a brief round-by-round summary of their decisions and outcomes, and are asked to provide in a blank space on their computer screens any relevant comments in particular about what might have aected their decisions during the experiment.

Subjects are also asked to solve the CRT test and a reduced-form Raven's test described above.

All Orsee [START_REF] Greiner | Subject pool recruitment procedures: organizing experiments with ORSEE[END_REF] and the experiment was computerized through software developed under Regate [START_REF] Zeiliger | A presentation of Regate, Internet based Software for Experimental Economics[END_REF] and z-Tree [START_REF] Fischbacher | z-Tree: Zurich toolbox for ready-made economic experiments[END_REF]. Sessions lasted about 45-60 minutes, with an average payo of roughly 18.83 euros (including a 5 euros show-up fee and 5 euros for completing the post-experimental tasks).

Results

Our main experimental results can be summarized as follows. First, in line with the existing literature, we observe systematic and sizable deviations from standard predictions based on the principle of dominance solvability. This phenomenon persists across game matrices and despite repetition. Second, we associate strategic behavior with cognitive skills. We nd that Raven's test score is a more reliable predictor of strategic behavior than CRT score: whenever the latter predicts behavior, the former does too, but not vice versa. Subjects with higher Raven's test scores are more likely to use the dominant strategy and to best respond to other player's dominant strategy.

Unlike those with low Raven's test score, they also react to the presence of strategic uncertainty. Note. Columns 1-10 summarize the frequencies of outcomes (dened in rows) as % of all outcomes observed in each round of a given experimental treatment. The last column provides overall results.

Aggregate behavior in experimental games

Table 4 outlines the main patterns of behavior in our experimental games. The statistical signicance of the changes observed in this table is tested by Models 1-3 in Table 5. We rst focus on the aggregate frequency of Pareto-Nash equilibrium (R, r) the sole outcome that survives the iterated elimination of (weakly) dominated strategies found in the Human condition. In both games, we observe substantial deviations from the predictions of this solution concept: overall, players attain the (R, r) outcome 58% of times in Game 1 and 43% in Game 2 (Model 1, H 0 : β 1 = 0, p = 0.318). We also observe that eciency increases over time: in both games, we observe the lowest frequency of (R, r) in the initial round (0.333 in Game 1 and 0.200 in Game 2), whereas the highest frequency of (R, r) occurs in the nal round (0.700 in Game 1 and 0.533 in Game 2).

To further explore the roots of these deviations, we turn to the aggregate patterns of both players' behavior in Human and Robot conditions. We focus on three behavioral dimensions of dominance solvability: the use of dominant strategy (captured by player Bs' behavior in the Human condition) and the ability to best respond to other player's dominant action with and without bearing the uncertainty about the latter (which is captured by player As' behavior in the Human and Robot conditions, respectively).

Ineciency is caused by both players, although their roles dier from one game to another: the scope of inecient behavior is similar for both players in Game 1, and highly asymmetric in Game 2. Overall, player As select action R with probability 0.730 in Game 1 and 0.447 in Game 2 (Model 3, H 0 : β 1 = 0, p = 0.047). However, player As' behavior happens to be misaligned with player and computed using the delete-one jackknife procedure. All models contain a dummy variable set to 1 for game matrix 2 (and 0 for game matrix 1). In Model 3, we also introduce an additional dummy variable set to 1 for Robot condition (and 0 for Human condition) and well as the interaction between these two variables. */**/*** indicate signicance at the 10%/5%/1% level.

Bs' actual decisions which follow the opposite trend: the total frequency of action r increases from 0.813 in Game 1 to 0.920 in Game 2 (Model 2, H 0 : β 1 = 0, p = 0.060). Importantly, the data from Robot sessions suggest that the uncertainty about player Bs' behavior is not the only driver of player As' choices. Player As frequently and systematically fail to best respond to player Bs' dominant action even when the latter comes with certainty in the Robot condition, although their willingness to select action R increases in both games as compared to the Human condition (to 0.773 in Game 1 and 0.690 in Game 2).

14 The fact that inecient actions from player As prevail in the absence of strategic uncertainty may suggest that at least some of them are boundedly rational decision makers.

In the next section, we analyze how these three behavioral components of dominance solvability vary as a function of players' cognitive skills.

Cognitive skills and strategic behavior

The average score in Raven's test (CRT) is 8.679 out of 16 with SD 3.117 (0.479 out of 3 with SD 0.852). Our experimental sample is properly randomized across treatments regarding both 14 Model 3 suggests that these two proportions are not signicantly dierent: testing H0 : β1 + β3 = 0 yields p = 0.303. The increase in the proportion of decisions R between Human and Robot conditions is insignicant for Game 1 (H0 : β2 = 0, p = 0.679) and signicant for Game 2 (H0 : β2 + β3 = 0, p = 0.054). measures. We do not reject the null hypothesis that Raven's test scores have the same distributions in all treatments (p = 0.275, Kruskal-Wallis test). A Kruskal-Wallis test applied to the CRT scores leads to the same conclusion (p = 0.502).

We also replicate several results from previous studies combining Raven's test and CRT regarding the relationship between both scores as well as gender dierences [START_REF] Brañas-Garza | Cognitive eort in the Beauty Contest Game[END_REF][START_REF] Corgnet | The cognitive basis of social behavior: cognitive reection overrides antisocial but not always prosocial motives[END_REF]. There is a moderate, yet highly signicant correlation between Raven and CRT scores (Spearman's ρ = 0.306, p < 0.001) which suggests that they may have a common source, but do not capture the same cognitive skills. Furthermore, the average score of males is signicantly higher than the average score of females (Raven's test: 9.382 with SD 0.341 vs 8.014 with SD 0.384, p = 0.009; CRT: 0.676 with SD 0.111 vs 0.291 with SD 0.087, p = 0.007; two-sided t-tests). 15

We also observe that many subjects (70%) of our 200 participants fail to provide at least one correct answer in our standard CRT. 16% provide exactly one , 8% two, and 6% three correct answers. This stands in line with Brañas-Garza, García-Muñoz, and Hernán González (2012) who report the respective frequencies of 67%, 23%, 9% and 1% for a similar sample size (N = 191), and echoes the scores in the least performant sample reported in a seminal study by [START_REF] Frederick | Cognitive Reection and Decision Making[END_REF]:

out of 138 students of the University of Toledo, 64% provide no correct answer, 21% provide one, 10% provide two, and 5% provide three corrects answers. In Figure 1, the sample is divided into two subsamples: subjects who provided at least one correct answer to CRT (referred to as CRT > 0) and those who did not (referred to as CRT = 0). The aggregate patterns of behavior weakly dier between the two subsamples. Bootstrap proportion tests fail to reject the null hypothesis that the overall proportions of decision R are the same for both CRT categories in the Human condition (p = 0.126) and in the Robot condition (p = 0.235). 16 The aggregate proportions of decision r, in turn, are found to be statistically dierent (p = 0.037), subjects with a CRT score zero being less likely to play r than subjects who gave at least one correct answer.

In Figure 2, we split our sample into three subsamples based on Raven's test score (1st tertile: less than 8 correct answers, 2nd tertile: between 8 and 10 correct answers, 3rd tertile: more than 15 See also [START_REF] Frederick | Cognitive Reection and Decision Making[END_REF] and Bosch-Domènech, Brañas-Garza, and Espín (2014) for related evidence.

16 We test the dierence in proportion of a given outcome between two experimental conditions by carrying out a bootstrap proportion test that accounts for within-subject correlation, i.e, the fact that the same individual takes 10 decisions. The procedure consists of bootstrapping subjects and their corresponding decisions over all ten rounds instead of bootstrapping decisions as independent observations (see, e.g., [START_REF] Jacquemet | Preference Elicitation under Oath[END_REF], for a detailed description of the procedure).

10 correct answers). Although bootstrap proportion tests suggest that player As' behavior in the Human condition does not vary signicantly between these three subsamples (1st tertile vs. 2nd

tertile: p = 0.255, 2nd vs. 3rd: p = 0.580, 1st vs 3rd: p = 0.565), signicant dierences arise for both player As in the Robot condition (p = 0.001, p = 0.735, p < 0.001, respectively) and for player Bs (p = 0.064, p = 0.057, p < 0.001, respectively). Raven's test score seems to have a more systematic association with players' behavior than CRT score, although both measures fail to predict behavior under strategic uncertainty.

Cognitive skills and dominance solvability: regression analysis

In what follows, we provide further econometric insights into these preliminary results. Following Brañas-Garza, García-Muñoz, and Hernán González (2012); Corgnet, Espín, and Hernáan-González (2015), we use three individual characteristics discussed in the previous section gender,

Raven's test score and CRT score (kept as a dummy variable with value 1 if the subject gave at least one correct asnwer at the CRT test and 0 otherwise) to explain behavior in our experimental games.

17 The econometric specication is based on the linear probability model and the estimation procedure is outlined in [START_REF] Jacquemet | What drives failure to maximize payos in the lab? A test of the inequality aversion hypothesis[END_REF]. We also control for payo scheme and repetition eects by including game matrix and round dummies. We consider three dierent outcome variables: player As' behavior in the Human and the Robot treatment, and player Bs' behavior in the Human treatment. Given the correlation between CRT and Raven's test scores, including both variables in the model might result in multicollinearity and lead to the under-rejection of the nullity of respective coecients. For each outcome, we rst include these two measures separately in Models 1 and 2, while Model 3 includes both variables. This evidence is summarized in Table 6.

We rst turn to player Bs' behavior. Models 1 and 2 suggest that both the coecient of CRT > 0 dummy and the coecient of Raven's test score are positive and signicant (p = 0.067

for CRT > 0 and p = 0.015 for Raven). In Model 3, the coecient of Raven's test score remains highly signicant (p = 0.014), while the coecient of CRT becomes insignicant (p = 0.253). Their joint signicance (p = 0.034) implies that cognitive skills predict the use of dominant strategy.

We now turn to player As' behavior in the Human condition. In order to formally test this conjecture, we now look at the reaction of player As with dierent cognitive skills to the disappearance of strategic uncertainty. Splitting the data according to Raven's score tertile, for each of the three subsamples we compare player As' behavior in the Human condition to their behavior in the Robot condition by regressing player As' choice on the Robot dummy (set to 1 for the Robot and to 0 for the Human condition). We also include the previous set of independent variables (except for Raven's test score itself ).

These results are summarized in Table 7. The coecient of the Robot dummy captures the eect of eliminating strategic uncertainty on player As' behavior for each of the three subsamples.

This suggests that only player As with high enough cognitive skills are sensitive to the uncertainty about player Bs' behavior. The behavior of players with low Raven's test score (1st tertile) is unresponsive to the degree of strategic uncertainty: the coecient of the Robot dummy is close to zero and insignicant (p = 0.822). For players with medium scores (2nd tertile), we nd a positive yet weakly signicant eect (p = 0.087) which becomes amplied and highly signicant for those player As whose Raven's test score belongs to the 3rd tertile of the experimental sample (p = 0.012).

Finally, it is also worth noting that player As' reaction to the payo scheme also varies as a function of Raven's test score. The coecient of the Game 2 dummy is close to zero and highly insignicant in the 1st tertile regression (p = 0.890). Then, it becomes negative in 2nd and 3rd tertile models (although it is only statistically signicant in the former with p = 0.012 and p = 0.271, respectively). This, in turn, stands in line with the previous nding that player As' willingness to play R increases as the safe choice L becomes less attractive relative to outcome (R, r). It also seems that the magnitude of this eect is mediated by player As' cognitive skills, although not in a monotone way.

Conclusion

This paper studies the relationship between strategic behavior and cognitive skills cognitive reection and uid intelligence in a classic 2 × 2 dominance-solvable game. Our results show that subjects with higher uid intelligence (measured by Raven's progressive matrices test) are more likely to play dominant strategy, and also more likely to best respond to other's strategy.

Furthermore, uid intelligence predicts strategic sophistication: only those players with suciently high Raven's test score are found to display sensitivity to the presence of uncertainty about others' behavior. Cognitive reection (measured by CRT), in turn, lacks the power to predict behavior in our experimental setting. We see three main conclusions that stem from these ndings.

First, these results contribute to the ongoing debate on the relationship between rationality and intelligence (see [START_REF] Stanovich | What Intelligence Tests Miss: The Psychology of Rational Thought[END_REF], for a critical review). For instance, [START_REF] Stanovich | What intelligence tests miss[END_REF] distinguish between two aspects of rational behavior: instrumental rationality which is understood as the ability to take appropriate action given one's goals and beliefs, and epistemic rationality which enables agents to hold beliefs that are commensurate with available evidence. In the strategic environment investigated in this paper, instrumental rationality can be associated with the ability to solve the game, while epistemic rationality with the ability to play it with others.

Our experimental data suggest an important relationship between uid intelligence (rather than reective thinking) and both of these facets of rationality in strategic settings. Both the ability to use dominance and iterated dominance to eciently solve the game, as well as the responsiveness to the availability of strategic information, is found to be predicted by Raven's test score (but not by CRT score).

The second contribution is related to the experimental methodology. Despite the fact that CRT and Raven's test are both commonly used to measure cognitive skills in experimental subject pools, still very little is known about their relative performance in predicting dierent types of behavior.

Therefore, the choice of one test over the other may happen to be at least as intuitive as evidencebased. As mentioned before, to the best of our knowledge only two experiments address this issue.

Brañas [START_REF] Brañas-Garza | Cognitive eort in the Beauty Contest Game[END_REF] do so in a strategic environment (pbeauty contest game), while [START_REF] Corgnet | The cognitive basis of social behavior: cognitive reection overrides antisocial but not always prosocial motives[END_REF] in a non-strategic one (individual choices on wealth distribution). Both studies nd that CRT performs better than Raven's test in predicting subjects' behavior. The result of the present experiment points the to the opposite conclusion. We believe that this dierence is driven by the very nature of the experimental tasks which may involve dierent types of cognitive eort. In our view, this issue deserves attention in future research.

Finally, although we nd evidence that behaving in accordance with dominance solvability is positively correlated with cognitive skills, we also substantiate that most of the variance in individual decision making cannot be explained by such skills. Thus, exploring factors alongside cognitive skills that generate strategic behavior remains an open and important empirical question. An interesting avenue is to disentangle individual determinants, e.g., personal characteristics (such as cognitive skills) that are associated with appropriate behavior, from environmental determinants, that is, those features of the decision making environment that lead decision makers to take certain types of actions.

  the sessions were conducted in February and March 2014. Out of the 200 participants (94 males), 155 were students with various elds of specialization. The majority of subjects (65%) had already taken part in economic experiments. Participants' average age was 25.6 (st. dev. is 7.5). All sessions took place at the Laboratoire d'Economie Experimentale de Paris (LEEP) at Paris School of Economics. Subjects were recruited via an on-line registration system based on

Figure 1 :

 1 Figure 1: CRT score and aggregate behavior across rounds and treatments

Figure 2 :

 2 Figure 2: Raven's test score and aggregate behavior across rounds and treatments

Table 1 :

 1 Generic form of the normal representation of[START_REF] Rosenthal | Games of perfect information, predatory pricing and the chain-store paradox[END_REF] dominance solvable game

		Player B
		l	r
	Player A	L (S ; s)	(S ; s)
		R (L ; m) (H ; h)

Table 2 :

 2 Overview of existing experimental evidence

	Experiment	Form		Payo			Outcomes (%)
			(L)	(R, r)	(R, l)	L R, r R, l r|R	r
	Beard, BeilTr.1	Seq	(9.75; 3.0)	(10; 5.0)	(3; 4.75)	66	29	6	83
	Beard, BeilTr.2	Seq	(9.00; 3.0)	(10; 5.0)	(3; 4.75)	65	35	0	100
	Beard, BeilTr.3	Seq	(7.00; 3.0)	(10; 5.0)	(3; 4.75)	20	80	0	100
	Beard, BeilTr.4	Seq	(9.75; 3.0)	(10; 5.0)	(3; 3.00)	47	53	0	100
	Beard, BeilTr.5	Seq	(9.75; 6.0)	(10; 5.0)	(3; 3.00)	86	14	0	100
	Beard, BeilTr.7	Seq	(58.50; 18.0)	(18.0; 28.50)	(60; 30.0)	67	33	0	100
	Beard et al.Tr.1	Seq	(1450; 450)	(1500; 750)	(450; 700)	79	18	3	83
	Beard et al.Tr.2	Seq	(1050; 450)	(1500; 750)	(450; 700)	50	32	18	64
	Goeree, HoltTr.1	Ext	(80; 50)	(90; 70)	(20; 10)	16	84	0	100
	Goeree, HoltTr.2	Ext	(80; 50)	(90; 70)	(20; 68)	52	36	12	75
	Goeree, HoltTr.3	Ext	(400; 250)	(450; 350)	(100; 348)	80	16	4	80
	Cooper, Van HuyckTr.9	Str	(4; 1)	(6; 5)	(2; 4)	27				86
	Cooper, Van HuyckTr.9	Ext	(4; 1)	(6; 5)	(2; 4)	21				84
	JZ, 2014BT1	Str	(9.75; 3.0)	(3.0; 4.75)	(10; 5.0)	51	41	8	84	81
	JZ, 2014ET1	Str	(9.75; 5.0)	(5.0; 9.75)	(10; 10.0)	54	33	13	72	73
	JZ, 2014ET3	Str	(9.75; 5.5)	(5.5; 8.50)	(10; 10.0)	39	48	13	79	76
	JZ, 2014ET4	Str	(8.50; 5.5)	(5.5; 8.50)	(10; 10.0)	25	61	14	82	82
	JZ, 2014ET2	Str	(8.50; 8.5)	(6.5; 8.50)	(10; 10.0)	26	70	4	94	94
	JZ, 2014BT2	Str	(8.50; 7.0)	(6.5; 7.00)	(10; 8.5)	26	70	4	94	94

Table 3

 3

			: The experimental games
		B			B
		l	r		l	r
		L (8.50 ; 3.00) ( 8.50 ; 3.00)		L (9.75 ; 8.50) ( 9.75 ; 8.50)
	A	R (6.50 ; 4.75) (10.00 ; 5.00)	A	R (3.00 ; 8.50) (10.00 ; 10.00)
		Game 1			Game 2
	2 Experimental design
	Our experiment is based on a 2 × 2 factorial design that varies the payo matrix and the nature of
	player B. Each of the four resulting experimental treatments is implemented through a between-
	subject procedure each subject participates in only one experimental condition. This data come
	from a large dataset, part of which has been previously used by

Table 4 :

 4 Aggregate results

						Round					Overall
		1	2	3	4	5	6	7	8	9	10
					P r(R, r) in the Human condition
	Game 1	0.333	0.600	0.667	0.700	0.567	0.600	0.433	0.633	0.567	0.700	0.580
	Game 2	0.200	0.333	0.400	0.400	0.433	0.500	0.500	0.433	0.467	0.533	0.420
				P r(r) by player B in the Human condition
	Game 1	0.767	0.800	0.867	0.900	0.800	0.800	0.700	0.833	0.867	0.800	0.813
	Game 2	0.833	0.933	0.900	0.933	1.000	0.933	0.933	0.900	0.900	0.933	0.920
				P r(R) by player A in the Human condition
	Game 1	0.500	0.733	0.700	0.767	0.767	0.800	0.700	0.767	0.700	0.867	0.730
	Game 2	0.300	0.333	0.400	0.400	0.433	0.533	0.533	0.500	0.500	0.533	0.447
				P r(R) by player A in the Robot condition
	Game 1	0.700	0.750	0.750	0.725	0.800	0.800	0.800	0.825	0.800	0.775	0.773
	Game 2	0.500	0.575	0.725	0.575	0.800	0.700	0.700	0.775	0.775	0.775	0.690

Table 5 :

 5 Aggregate results: statistical support

		Model 1	Model 2	Model 3
		P r(R, r)	P r(r)	P r(R)
	Constant (β 0 )	0.580***	0.813***	0.730***
		(0.144)	(0.032)	(0.084)
	1[Game 2] (β 1 )	-0.160	0.107*	-0.283**
		(0.103)	(0.044)	(0.140)
	1[Robot] (β 2 )			0.043
				(0.103)
	1[Robot] × 1[Game 2] (β 3 )			0.201
				(0.161)
	N	600	600	1400
	R 2	0.026	0.025	0.066

Note. Estimates of linear probability models on outcome (R, r) (Model 1), decision r by player B (Model 2) and decision R by player A (Model 3). Standard errors (in parentheses) are clustered at the session level in Human treatments (3 clusters per game matrix, 6 in total) and individual level in the Robot condition (40 clusters per game matrix, 80 in total)

  Notwithstanding the previous set of results, cognitive skills are not found to explain player As' choices. The coecient of CRT > 0 dummy is insignicant (p = 0.226) in Model 1, and so is the coecient of Raven's test score (p = 0.633) in Model 2. If we account for both, Model 3 reveals that the coecients of both scores are neither individually (p = 0.226 for CRT > 0 and p = 0.550 for Raven's test score) nor jointly signicant (p = 0.503). Finally, the behavior of player As in the Robot condition is only predicted by Raven's test score: unlike CRT > 0 dummy, its coecient remains positive

Table 6 :

 6 Cognitive predictors of strategic behavior: regression analysis Estimates of linear probability models explaining the likelihood of decision R by player A and decision r by player B. Standard errors (in parantheses) are clustered at the session level in the Human condition (3 clusters per game matrix, and highly signicant across models (p ≤ 0.001). Unsurprisingly, the joint insignicance of both coecients in Model 3 is also rejected (p = 0.003). , the results presented in Table6suggest that cognitive skills predict certain components of strategic behavior: the use of dominant strategy (reected in player Bs' behavior), as well as the ability to best respond to other player's dominant strategy (reected in player As' behavior in the Robot condition). Moreover, in both cases Raven's test score is a more reliable predictor of behavior than CRT score. However, we also observe that Raven's test score fails to predict player As' behavior once player Bs' behavior becomes uncertain, that is once we move from Robot to Human condition. This, in turn, points towards an interplay between the degree of strategic uncertainty, behavior in the experimental games, and individual cognitive skills. Importantly, the existence of such an interplay is also supported by Figure2which shows that the aggregate levels of eciency shift upwards between the Human condition and the Robot condition for the 2nd and 3rd Raven's score tertile, but not the 1st tertile.

				Pr(R) by player A			Pr(r) by player B
		Human condition	Robot condition		Human condition
		Model 1 Model 2 Model 3 Model 1	Model 2	Model 3	Model 1 Model 2 Model 3
	Const.	0.423*** 0.552** 0.563** 0.573***	0.242*	0.240*	0.705*** 0.430*** 0.444***
		(0.080)	(0.176)	(0.197)	(0.088)	(0.135)	(0.135)	(0.027)	(0.103)	(0.099)
	1[CRT>0]	0.131		0.152	0.062		(0.024)	0.109*		0.046
		(0.095)		(0.121)	(0.102)		(0.102)	(0.047)		(0.036)
	Raven		0.013	0.018		0.0426*** 0.0434***		0.0313** 0.0287**
			(0.025)	(0.027)		(0.013)	(0.012)		(0.009)	(0.008)
	1[Game 2]	-0.270*	0.263	0.266	0.068	0.054	0.056	0.100	0.132*	0.129*
		(0.129)	(0.139)	(0.136)	(0.083)	(0.076)	(0.079)	(0.052)	(0.056)	(0.056)
	1[Male]	0.132	0.187	0.158	0.096	0.072	0.077	0.025	0.024	0.017
		(0.126)	(0.100)	(0.107)	(0.090)	(0.076)	(0.089)	(0.046)	(0.046)	(0.047)
	Round:
	2	0.133	0.133	0.133	0.063	0.063	0.063	0.067	0.067	0.067
		(0.092)	(0.092)	(0.092)	(0.048)	(0.048)	(0.048)	(0.042)	(0.042)	(0.042)
	3	0.150	0.150	0.150	0.138*** 0.138***	0.138***	0.083	0.083	0.083
		(0.109)	(0.109)	(0.109)	(0.050)	(0.050)	(0.050)	(0.048)	(0.048)	(0.048)
	4	0.183**	0.183** 0.183**	0.050	0.050	0.050	0.117*	0.117*	0.117*
		(0.070)	(0.070)	(0.070)	(0.047)	(0.047)	(0.047)	(0.048)	(0.048)	(0.048)
	5	0.200*	0.200*	0.200*	0.200*** 0.200***	0.200***	0.100	0.100	0.100
		(0.089)	(0.089)	(0.089)	(0.045)	(0.045)	(0.045)	(0.052)	(0.052)	(0.052)
	6	0.267**	0.267** 0.267** 0.150*** 0.150***	0.150***	0.067	0.067	0.067
		(0.088)	(0.088)	(0.088)	(0.054)	(0.054)	(0.054)	(0.049)	(0.049)	(0.049)
	7	0.217*	0.217*	0.217*	0.150*** 0.150***	0.150***	0.017	0.017	0.017
		(0.098)	(0.098)	(0.098)	(0.047)	(0.047)	(0.047)	(0.048)	(0.048)	(0.048)
	8	0.233**	0.233** 0.233** 0.200*** 0.200***	0.200***	0.067	0.067	0.067
		(0.088)	(0.088)	(0.088)	(0.057)	(0.057)	(0.057)	(0.056)	(0.056)	(0.056)
	9	0.200	0.200	0.200	0.188*** 0.188***	0.188***	0.083	0.083	0.083
		(0.113)	(0.113)	(0.113)	(0.047)	(0.047)	(0.047)	(0.060)	(0.060)	(0.060)
	10	0.300**	0.300** 0.300** 0.175*** 0.175***	0.175***	0.067	0.067	0.067
		(0.115)	(0.115)	(0.115)	(0.056)	(0.056)	(0.056)	(0.049)	(0.049)	(0.049)
	R 2	0.151	0.141	0.160	0.050	0.139	0.140	0.060	0.108	0.111
	Note.

Table 7 :

 7 The effect of strategic uncertainty and cognitive skills: evidence from player As' . Estimates of linear probability models on decision R by player A. Standard errors (in parentheses) are clustered at the session level in the Human condition (3 clusters per game matrix, 6 in total) and individual level in the Robot condition (40 clusters per game matrix, 80 in total) and computed using the delete-one jackknife procedure. Data from Human and Robot conditions are pooled and split into three subsamples based on Raven's test score tertiles. Other independent variables include a dummy set to 1 for a positive CRT score, as well as gender, game matrix and round dummies (omitted from the table). */**/*** indicate signicance at the 10%/5%/1% level.

	behavior in Human and Robot conditions
		Raven's test score tertile
		1st	2nd	3rd
	Constant	0.277*	0.592***	0.330**
		(0.147)	(0.065)	(0.135)
	1[Robot]	0.044	0.158*	0.428**
		(0.195)	(0.090)	(0.155)
	1[CRT>0]	0.002	0.038	0.016
		(0.262)	(0.066)	(0.188)
	1[Male]	0.144	0.033	0.212
		(0.145)	(0.063)	(0.179)
	1[Game 2]	0.034	-0.245**	-0.176
		(0.146)	(0.092)	(0.155)
	Round dummies	Yes	Yes	Yes
	N	480	610	310
	R 2	0.048	0.173	0.298

Note

This class of games has been rst introduced by[START_REF] Moulin | Game Theory for the Social Sciences[END_REF] as the p-beauty contest games, where p (often equal

2/3) stands for the target fraction of all numbers' average.2 See[START_REF] Nagel | Unraveling in Guessing Games: An Experimental Study[END_REF] and[START_REF] Ho | Iterated Dominance and Iterated Best Response in Experimental p-Beauty Contests[END_REF] for early evidence from the laboratory, Costa-Gomes and[START_REF] Costa-Gomes | Cognition and Behavior in Two-Person Guessing Games: An Experimental Study[END_REF] for a laboratory experiment supporting a behavioral model of bounded rationality, and Bosch-Domenech, Montalvo,[START_REF] Bosch-Domenech | Newspaper and lab beauty-contest experiments[END_REF] for related evidence from the eld.

The main conclusion that can be drawn from these studies is that high cognitive skills predict strategic sophistication and ecient decision making. First, people with high cognitive skills make more accurate predictions about other people's intentions. Recent evidence from psychological research reveals the relationship between cognitive skills and the theory of mind. Using the Reading the Mind in the Eyes test (RMET,[START_REF] Baron-Cohen | The Reading the Mind in the Eyes" test revised version: A study with normal adults, and adults with Asperger syndrome or high-functioning autism[END_REF] to measure one's theory of mind, Ibanez, Huepe, Gempp, Gutiérrez, Rivera-Rei, and Toledo (2013) nd that people with higher cognitive skills are better at infering the internal emotional states of

others. 6 Relatedly, the results of a neuroeconomic experiment on 5 Cognitive skills are often measured using (amongst others) the Cognitive Reection Test(CRT, Frederick, 2005), the Raven's progressive matrices test(Raven, 2008), or both (like in this study). The details of these two measures are presented in Section 2.6 RMET consists of a series of photos of the area of the face involving the eyes. Subjects are asked to choose one of the four words that best describes what the person in the photo is thinking or feeling.

[START_REF] Al-Ubaydli | Patience, cognitive skill, and coordination in the repeated stag hunt[END_REF][START_REF] Al-Ubaydli | Patience, cognitive skill, and coordination in the repeated stag hunt[END_REF] also report that individual cognitive skills do not predict individual willingness to reach ecient outcomes in these two game.

Given that most CRT scores in our sample are null and the higher the score, the less frequent it gets, di- chotomizing the CRT score variable limits the impact of the outliers on the overall results.

in total) and individual level in the Robot condition (40 clusters per game matrix, 80 in total) and computed using the delete-one jackknife procedure. Models 1 and 2 include a single measure of cognitive skills (a dummy set to 1 for a positive CRT score, or Raven's test score), while Model 3 combines both variables. Other independent variables include gender, game matrix and round dummies. The number of observations is N = 600 for Human and N = 800 for Robot conditions. */**/*** indicate signicance at the 10%/5%/1% level.

* This project has received funding from JSPS-ANR bilateral research grant BECOA (ANR-11-FRJA-0002), as well as the LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, and LABEX OSE of the Paris School of Economics (ANR-10-LABX_93-01), both within the program Investissements d'Avenir" (ANR-11-IDEX-007) operated by the French National Research Agency (ANR). Ivan Ouss provided ecient research assistance.


Hanna Becker
email: hannabecker6@hotmail.com

Laurent Albera

Pierre Comon

Rémi Gribonval

Fabrice Wendling

Isabelle Merlet

Localization of distributed EEG sources in the context of epilepsy: a simulation study

Keywords: EEG, distributed source localization, epilepsy, brain source imaging

Objective: Surface EEG recordings are routinely performed for the diagnosis and management of epilepsy. More particularly, they can help to delineate the brain regions involved in interictal epileptic activity. This is achieved by applying distributed source localization algorithms to the EEG data. Over the last two decades, a multitude of different methods have been proposed. The objective of this paper consists in comparing the performance of eight representative algorithms taking into account recently developed methods. Methods: The performance comparison is based on realistic computer simulations in the context of epileptic source localization. Main findings: All tested algorithms generally permit to identify the source positions, but only some algorithms (STWV-DA, 4-ExSo-MUSIC, SVB-SCCD, cLORETA) also give an indication of the spatial extent of the sources. Localizing deep and mesial sources remains a challenging problem.

Introduction

Surface ElectroEncephaloGraphy (EEG) recordings permit to collect valuable information about the cerebral sources that are at the origin of the observed electromagnetic brain activity. This information is crucial for the diagnosis and management of some diseases. In this paper, we focus in particular on the application of EEG in the context of epilepsy, where the objective consists in identifying the brain regions which are involved in epileptic activity between seizures. The precise delineation of these areas is essential for the evaluation of patients with drug-resistant partial epilepsy for whom a surgical intervention can be considered to remove the epileptogenic brain regions that are responsible for the occurrence of seizures. To this end, source localization techniques are applied to interictal spikes as has been done in several previous studies [START_REF] Ebersole | Noninvasive localization of epileptogenic foci by EEG source modeling[END_REF][START_REF] Merlet | Dipole modeling of interictal and ictal EEG and MEG[END_REF][START_REF] Michel | EEG source imaging[END_REF][START_REF] Gavaret | Electric source imaging in frontal lobe epilepsy[END_REF][START_REF] Plummer | EEG source localization in focal epilepsy: Where are we now?[END_REF][START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF]. Most of these studies have focused on identifying equivalent current dipoles, each of which summarizes the epileptic activity in a certain brain area. Yet it has been shown that the epileptic paroxysms which are observable by EEG measurements often involve large cortical regions [START_REF] Cosandier-Rimélé | The neuronal sources of EEG: Modeling of simultaneous scalp and intracerebral recordings in epilepsy[END_REF][START_REF] Gavaret | Electric source imaging in frontal lobe epilepsy[END_REF][START_REF] Tao | Intracranial EEG substrates of scalp EEG interictal spikes[END_REF][START_REF] Merlet | Reliability of dipole models of epileptic spikes[END_REF][START_REF] Ebersole | Magnetoencephalography/magnetic source imaging in the assessment of patients with epilepsy[END_REF]. Therefore, our goal is not only to identify the locations of the epileptic regions, but also to determine their spatial extents. This is achieved by localizing distributed sources, which are characterized by a high number of dipoles (largely exceeding the number of sensors) with fixed locations.

Localizing distributed brain sources from the noisy mixture of signals which is recorded by the EEG sensors is also referred to as brain source imaging and requires solving an ill-posed inverse problem. In order to restore identifiability, additional hypotheses about the sources have to be made. Over the past two decades, a multitude of brain source imaging algorithms have been proposed (see, e.g., [START_REF] Baillet | Electromagnetic brain mapping[END_REF][START_REF] Grech | Review on solving the inverse problem in EEG source analysis[END_REF][START_REF] Wipf | A unified Bayesian framework for MEG/EEG source imaging[END_REF][START_REF] Becker | Brain source imaging: from sparse to tensor models[END_REF]), that differ essentially in the exploited hypotheses as well as in methodological aspects. In [START_REF] Becker | Brain source imaging: from sparse to tensor models[END_REF], these approaches have been dinstinguished into four groups: regularized least squares, Bayesian, tensor-based, and extended source scanning methods. Even though several studies [START_REF] Pascual-Marqui | Review of methods for solving the EEG inverse problem[END_REF][START_REF] Grova | Evaluation of EEG localization methods using realistic simulations of interictal spikes[END_REF][START_REF] Grech | Review on solving the inverse problem in EEG source analysis[END_REF][START_REF] Chowdhury | MEG source localization of spatially extended generators for epileptic activity: comparing entropic and hierarchical Bayesian approaches[END_REF][START_REF] Becker | Brain source imaging: from sparse to tensor models[END_REF] have aimed at comparing different source imaging algorithms, a full and deep study of these methods that takes into account recent advances in this field is still missing. The objective of this paper is to conduct a thorough, comparative simulation study with eight representative algorithms belonging to different classes of source imaging approaches. This study extends the preliminary results reported in [START_REF] Becker | Brain source imaging: from sparse to tensor models[END_REF] by considering a significantly wider panel of scenarios and an updated list of algorithms.

Materials and methods

Data model

The brain electric and magnetic fields are generated by the current flows that result from the electrochemical process of information transmission between neurons. To obtain a signal of sufficient amplitude to be measurable at the surface of the scalp, a certain number of simultaneously active neuronal populations is required. These populations can be modeled by a number of current dipoles belonging to a pre-defined source space. As the EEG recordings are mostly generated by pyramidal cells located in the gray matter with an orientation perpendicular to the cortex [START_REF] Dale | Improved localization of cortical activity by combining EEG and MEG with MRI cortical surface reconstruction: a linear approach[END_REF], it is physiologically plausible to assume that the D dipoles of the source space are located on the cortical surface with fixed orientations perpendicular to this surface. The electric potential data that is recorded at the N electrodes of an EEG sensor array for T time samples then constitutes the superposition of all source dipole signals contained in the signal matrix S ∈ R D×T that are transmitted to the surface of the scalp. The propagation of the signals in the head volume conductor is characterized by the lead field matrix G ∈ R N ×D . This leads to the following model for the EEG data:

X = GS. (1)
In the context of epilepsy, the epileptic regions can be modeled by distributed sources that can be described as the union of (one or) several contiguous areas of the cortex (so-called patches) with highly correlated source activities [START_REF] Cosandier-Rimélé | A physiologically plausible spatio-temporal model for EEG signals recorded with intracerebral electrodes in human partial epilepsy[END_REF][START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF]. All dipoles that do not belong to a distributed source can be considered to emit a background activity. Consequently, to distinguish between the P distributed sources, that we want to retrieve, and the noisy background activity, we can rewrite the data model (1) in the following way:

X = P p=1 kp∈Ωp g kp s T kp + l / ∈∪ P p=1 Ωp g l s T l = X e + X b . (2)
Here, Ω p is the set of indices of the dipoles that belong to the p-th distributed source, g k is the lead field vector of the k-th dipole, and s k is the associated signal vector that corresponds to the k-th row vector of S. The matrices X e and X b contain the epileptic and background activity, respectively.

For a given head model and source space, the lead field matrix G can be computed numerically [START_REF] Gramfort | Mapping, timing and tracking cortical activations with MEG and EEG: Methods and application to human vision[END_REF]. Therefore, we assume in this paper that it is known. The objective of brain source imaging then consists in estimating the unknown sources S from the measurements X. As the number of source dipoles D (several thousands) is much higher than the number of sensors (several hundreds), the lead field matrix is severely underdetermined, making the inverse problem ill-posed. In order to restore identifiability, additional hypotheses about the sources have to be made, giving rise to a large number of different source imaging algorithms.

Algorithms

In this paper, we concentrate on the following eight source imaging algorithms: sLORETA, cLORETA, MCE, SVB-SCCD, MxNE, Champagne, 4-ExSo-MUSIC, and STWV-DA.

The first five methods belong to the class of regularized least squares approaches. More particularly, sLORETA [START_REF] Pascual-Marqui | Standardized low resolution brain electromagnetic tomography (sLORETA): technical details[END_REF] employs an L2-norm regularization strategy like the classical minimum norm estimate and normalizes the solution with respect to the estimated noise level. cLORETA [START_REF] Wagner | Smooth reconstruction of cortical sources from EEG and MEG recordings[END_REF] aims at identifying a spatially smooth source distribution by applying a Laplacian operator L to the source matrix in the L2-norm regularization term.

MCE [START_REF] Uutela | Visualization of magnetoencephalographic data using minimum current estimates[END_REF] identifies a sparse source estimate by replacing the L2-norm in the regularization term by an L1-norm. SVB-SCCD [START_REF] Becker | Fast, variationbased methods for the analysis of extended brain sources[END_REF] leads to a piece-wise constant source estimate, which is obtained by employing L1-norm regularization both in the original source domain and on the variational map, which characterizes the changes in amplitude between adjacent source dipoles. Finally, the MxNE algorithm [START_REF] Ou | A distributed spatio-temporal EEG/MEG inverse solver[END_REF] assumes that the active source dipoles do not change over the considered time interval, which is imposed using a mixed norm (L12-norm) regularization strategy.

The Champagne algorithm [START_REF] Wipf | Robust Bayesian estimation of the location, orientation, and time course of multiple correlated neural sources using MEG[END_REF] is a representative of empirical Bayesian methods, which are based on a probabilistic model of the data. Assuming that the sources at each time point are described by a zero-mean Gaussian random vector with independent elements, the goal of the algorithm is to obtain an estimate for the source covariance matrix, which can then be used to infer an estimate of the sources.

4-ExSo-MUSIC [START_REF] Birot | Localisation of extended brain sources from EEG/MEG: the ExSo-MUSIC approach[END_REF] is an extended source scanning method, which is based on the quadricovariance matrix containing the fourth order cumulants estimated from the data. More particularly, it extends the classical MUSIC algorithm, which has previously been used for equivalent dipole localization, to the identification of distributed sources based on higher order statistics by employing a dictionary of circular-shaped source regions of varying positions and sizes, the so-called disks.

STWV-DA [START_REF] Becker | EEG extended source localization: tensor-based vs. conventional methods[END_REF] is a tensor-based source imaging algorithm, which proceeds in two steps. The first step consists in separating several simultaneously active distributed sources based on tensor decomposition of space-time-wavevector data. In the second step, the results of the tensor decomposition are used for source localization, which is performed for each distributed source separately based on a dictionary of disks similar to 4-ExSo-MUSIC. To generate physiologically plausible brain signals, a model based on coupled neuronal populations [START_REF] Cosandier-Rimélé | A physiologically plausible spatio-temporal model for EEG signals recorded with intracerebral electrodes in human partial epilepsy[END_REF][START_REF] Cosandier-Rimélé | The neuronal sources of EEG: Modeling of simultaneous scalp and intracerebral recordings in epilepsy[END_REF] is employed. This model is used to simulate interictal epileptiform spike-like signals for the source dipoles belonging to the patches and normal background activity of the brain for all other dipoles.

Tested methods

All source imaging algorithms are applied to spatially prewhitened data, except for Champagne, which directly uses the noise covariance instead. The noise covariance matrix is estimated using 25000 time samples of data generated for the case where all dipoles emit background activity. To improve the SNR, we average the data of 10 spikes. Since sLORETA, cLORETA, MCE, and SVB-SCCD do not take into account the temporal information, they are applied to the time sample corresponding to the maximum of the epileptiform spike. STWV-DA, MxNE and Champagne are applied to the 200 time samples of averaged epileptic activity. For 4-ExSo-MUSIC, to have a sufficient amount of time samples at our disposal to accurately estimate the fourth order statistics, we concatenate the 200 time samples that are selected for each of the 10 spikes, leading to a total of 2000 time samples.

The algorithms are implemented as follows: for sLORETA and cLORETA, the regularization parameter is fixed such that it approximately balances between the data fit and the regularization term. MCE and MxNE are implemented using FISTA [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]. The regularization parameter is chosen according to the level of sparsity that we aim to achieve. SVB-SCCD is implemented as described in [START_REF] Becker | Fast, variationbased methods for the analysis of extended brain sources[END_REF]. The implementation of Champagne follows [START_REF] Wipf | Robust Bayesian estimation of the location, orientation, and time course of multiple correlated neural sources using MEG[END_REF], but has been modified to take into account the fixed orientations of the source dipoles.

For STWV-DA, the tensor is constructed and decomposed as described in [START_REF] Becker | EEG extended source localization: tensor-based vs. conventional methods[END_REF] with a number of components that is equal to the number of patches.

For 4-ExSo-MUSIC, the dimension of the signal subspace is chosen according to the number of patches. For both STWV-DA and 4-ExSo-MUSIC, we construct a dictionary of disks comprising up to 100 grid dipoles.

Evaluation criteria

To quantitatively evaluate the performance of the algorithms, we use two measures1 : the Distance of Localization Error (DLE) [START_REF] Yao | Evaluation of different cortical source localization methods using simulated and experimental EEG data[END_REF], which characterizes the difference between the original and the estimated source configurations, and the Receiver Operating Characteristic (ROC) [START_REF] Grova | Evaluation of EEG localization methods using realistic simulations of interictal spikes[END_REF], which reflects the ability of a source imaging algorithm to recover the extent and the form of the distributed sources.

Mathematically, the DLE is defined as follows:

DLE = 1 2   1 Q k∈I min ∈ Î ||r k -r || 2 + 1 Q ∈ Î min k∈I ||r k -r || 2   (3
TPF = #(I ∩ Î) #I ; FPF = # Î -#(I ∩ Î) #J -#I . (4)
Here, J denotes the set of all dipoles belonging to the source space. Different Concerning the patch location, the examined source imaging algorithms generally yield good results for superficial patches (InfFr, InfPa, SupOcc, and MidTe) with exception of patch PreC. However, the algorithms exhibit some difficulties for accurately recovering deep patches such as BasTe and Hipp, as well as for the patch Cing, located between the two hemispheres.

Influence of the patch size

To determine the influence of the source extent on the performance of the different source imaging algorithms, we vary the size of the patch SupFr

Influence of the patch number

In practice, one is often confronted with measurements that originate from several quasi-simultaneous active source regions within the brain. In this section, we analyze the ability of the source imaging algorithms to identify two or three patches that are involved in epileptic spike propagation.

Discussion

In summary, for the considered scenarios, sLORETA, Champagne, MCE, and MxNE recover well the source positions, though not their spatial extents as they are conceived for focal sources, while 4-ExSo-MUSIC, STWV-DA, and SVB-SCCD also permit to obtain an accurate estimate of the source size.

cLORETA leads to intermediate results as the identified dipoles with high amplitudes and smooth source distributions correspond well to the patches, but make it difficult to delineate the source regions. Nevertheless, the spatial smoothness prior has proven to be especially effective for large patch sizes where cLORETA achieves the best performance in terms of DLE.

Among the methods for focal source reconstruction, Champagne leads to the sparsest source distributions, identifying only a very small number of dipoles. This result might be explained by the fact that Champagne is based on the assumption that all grid dipoles emit independent source activities, which is violated by the patches. Moreover, in the Champagne algorithm, there is no parameter that can be adjusted to obtain different levels of sparsity, contrary to MCE and MxNE where this is achieved by varying the regularization parameter. At the same time, this is also an important advantage of Champagne because in practice, the tuning of parameters is tedious and time-consuming, and even though Champagne does not identify the source extents, it still permits to accurately localize the foci of the source activity. Combined with an adequate scheme for distributed source localization, Champagne could thus become a powerful tool for source imaging.

While STWV-DA and 4-ExSo-MUSIC lead to similar patch estimates for the single patch scenarios and in some cases also for multipatch scenarios, all in all, STWV-DA outperforms 4-ExSo-MUSIC as it leads to better source estimates in the presence of multiple patches, where 4-ExSo-MUSIC does not localize all patches or leads to erroneous patch estimates. This problem is due to the limitation of the employed dictionary to distributed sources composed of one disk and could be overcome by modifying the dictionary to include combinations of several disks. However, this has not been done so far. Compared to SVB-SCCD, STWV-DA only leads to better results for the smallest considered patch. Otherwise, SVB-SCCD yields slightly better source estimates than STWV-DA, which is due to its greater flexibility in recovering the patch shape. Because of the use of a dictionary of disks, STWV-DA tends to recover circular-shaped source regions.

Even though this has not been explicitly shown in the above simulations, we noticed that most of the methods except for STWV-DA require prewhitening of the data or a good estimate of the noise covariance matrix (in case of Champagne) in order to yield accurate results. On the one hand, this can be explained by the assumption of spatially white Gaussian noise made by some approaches, while on the other hand, the prewhitening also leads to a decorrelation of the rows of the lead field matrix and therefore to a better conditioning of the lead field matrix, which consequently facilitates the correct identification of active grid dipoles.

Conclusion

We have compared eight source imaging algorithms using realistic simulated EEG data in the context of epilepsy. We have observed that these methods generally work well for superficial patches, but have difficulties in identifying deep patches and mesial sources, located close to the midline.

While all algorithms generally permit to identify the source positions, only STWV-DA, 4-ExSo-MUSIC, SVB-SCCD, and, to some extent, cLORETA, also give an indication of the spatial extent of the sources. On the whole, for the situations addressed in our simulation study, STWV-DA and SVB-SCCD seem to be the most promising algorithms for distributed source localization.

Figure captions and tables

Illustration of a realistic head model with three compartments representing the brain, the skull, and the scalp and a source space that consists of a large number of dipoles (represented by black dots) located on the gray matter/white matter interface.

Figure 2

Location of the 11 patches that are considered for the simulations in this paper. We provide 6 different views of the cortical surface such that all patches can be seen. For convenience, we refer to these patches using the following names that indicate the patch positions: SupFr -superior frontal gyrus, InfFr -inferior frontal gyrus, PreC -precentral gyrus, SupTe -superior temporal gyrus, MidTe -middle temporal gyrus, BasTe -basal aspect of the temporal lobe, OccTe -occipital temporal junction, InfPa -inferior parietal lobule, SupOcc -superior occipital gyrus, Cing -cingulate gyrus, Hipp -para-hippocampal gyrus. We provide 6 different views of the cortical surface such that all patches can be seen.

For convenience, we refer to these patches using the following names that indicate the patch positions: SupFr -superior frontal gyrus, InfFr -inferior frontal gyrus, PreCprecentral gyrus, SupTe -superior temporal gyrus, MidTe -middle temporal gyrus, BasTe -basal aspect of the temporal lobe, OccTe -occipital temporal junction, InfPa -inferior parietal lobule, SupOcc -superior occipital gyrus, Cing -cingulate gyrus, Hipp -parahippocampal gyrus.

2. 3 .

 3 Simulation setup 2.3.1. Data generationWe simulate physiologically plausible EEG recordings for an electrode cap comprising N = 91 electrodes and data intervals with a length of T = 200 time samples at a sampling frequency of 256 Hz. To this end, we consider a realistic head model composed of three compartments representing the brain, the skull, and the scalp (see Fig.1), which have been derived from a single subject MRI. The source space is defined by the triangularized inner cortical surface (gray matter / white matter interface), segmented from the same MRI. A grid dipole is placed at the centroid of each of the triangles with an orientation perpendicular to the triangle's surface. The grid consists of 19626 triangles and on average, each triangle describes 5 mm 2 of the cortical surface. The lead field matrix G is then computed numerically using a boundary element method (ASA, ENT, Enschede, The Netherlands). For the generation of distributed sources, we consider 11 different patches located on the left hemisphere (see Fig.2). Each patch consists of 100 adjacent grid dipoles corresponding to a cortical area of approximately 5 cm 2 .

  )where I denotes the set of indices of all grid dipoles belonging to an active patch, Q is the number of active grid dipoles, i.e., Q = #I, Î denotes the set of indices of all estimated active source dipoles and Q = # Î corresponds to the number of the latter. Furthermore, r k denotes the position of the k-th source dipole. To compare the estimated source configuration to the original source configuration characterized by the dipoles belonging to the active patches, we consider a number of active estimated dipoles that is equal to the true number of patch dipoles or as close to this number as possible. To achieve this, we threshold the absolute value of the sLORETA, cLORETA, and SVB-SCCD solutions and the STWV-DA, and 4-ExSo-MUSIC metrics by a suitable value. In case of MCE and MxNE, we choose the regularization parameter to obtain a number of non-zero elements that lies between 80 % and 100 % of the number of active sources.The ROC displays the True Positive Fraction (TPF) of correctly identified source dipoles as a function of the False Positive Fraction (FPF), which represents the number of source dipoles erroneously associated with the distributed sources:

  TPF and FPF values are achieved by varying the threshold values for the extended source localization algorithms. The results are averaged over 50 realizations with different spike-like signals and varying background activity.

3 . Results 3 . 1 .

 331 Influence of the patch position Superficial sources exhibit more focal distributions of the electric potential than deep sources. Furthermore, the signals emanating from deep sources lead to smaller amplitudes at the sensor level (and thereby, smaller SNRs) than those of superficial sources. It is thus significant to determine the influence of the patch position on the localization accuracy of the different source imaging methods. To this end, we consider 8 different patches: InfFr, InfPa, Cing, SupOcc, PreC, BasTe, MidTe, and Hipp. The results in terms of ROC curves and DLE are shown in Fig.3and Table1, respectively. The patches that are recovered by the algorithm which yields the highest TPF at an FPF of 0.2 % are shown in Fig.4in comparison to the original patches.For patches InfPa, SupOcc, BasTe, and MidTe, the best results in terms of both DLE and ROC are achieved by SVB-SCCD, whereas STWV-DA and 4-ExSo-MUSIC yield the best performance for patches InfFr, Cing, PreC, and Hipp. These two methods lead to almost identical source imaging results because in the case of single patches, there is no source separation to be performed by STWV-DA and the patch estimates are mostly influenced by the employed dictionary of disks, which is the same for STWV-DA and 4-ExSo-MUSIC. sLORETA and cLORETA generally yield better results in terms of both DLE and ROC than MCE and MxNE. Surprisingly, MCE slightly outperforms MxNE for all tested scenarios, whereas one could have expected that the exploitation of temporal information would lead to more robust source estimates. However, this result could be related to our somewhat abusive use of the sparse source estimation techniques to recover patches of larger extent. Champagne leads to intermediate DLEs because it always identifies a small number of dipoles at the correct source position. Albeit, as the ROC curves show, this method is the least suited to recover the spatial extent of the patches because it recovers very sparse source estimates.

  and consider three patches composed of 10, 100, and 400 grid dipoles corresponding to an area of approximately 50, 500 and 2000 mm 2 . The simulation results in terms of ROC curves, estimated source reconstructions, and DLE values are shown in Fig.s 5 and 6, and Table 2, respectively. The performance of sLORETA, Champagne, MCE, and MxNE in terms of ROC decreases as the patch area increases. This is due to the fact that these methods, especially MCE and MxNE, are conceived for focal sources and are not well suited to recover the spatial extent of the sources. 4-ExSo-MUSIC, STWV-DA, and SVB-SCCD on the other hand have been developed to localize distributed sources and can therefore accurately recover sources of larger size. The hypothesis of spatial smoothness exploited by cLORETA also favors the reconstruction of extended sources, leading to the best performance among all tested methods for the largest considered patch.In terms of DLE, all methods except for cLORETA yield the highest reconstruction accuracy for a patch of medium size. For cLORETA, we observe a continually decreasing DLE for augmenting patch size. For the smallest patch, the best source reconstructions are achieved by STWV-DA and 4-ExSo-MUSIC. The source dipoles of maximal amplitudes identified by Champagne, MCE, and MxNE are close to the true patch, but not exactly at the correct position. Furthermore, SVB-SCCD overestimates the size of the patch, recovering a much larger source region, as is the case for sLORETA and cLORETA. In fact, these three methods yield nearly the same estimated source distributions for the patches of small and medium sizes. These source reconstructions are better suited for the medium-sized patch, which leads to smaller DLE in this case. In particular, SVB-SCCD achieves the best DLE of all methods for the medium-sized patch. Note that sLORETA, cLORETA, MCE, and MxNE also identify several dipoles on the right hemisphere, unlike the distributed source localization algorithms SVB-SCCD, STWV-DA, and 4-ExSo-MUSIC. Nevertheless, for cLORETA, the amplitudes of the contralateral source dipoles are rather small. For the largest patch, cLORETA and SVB-SCCD outperform all other tested methods. While the sLORETA solution indicates also a patch of larger extent, the sparse estimates obtained by MCE, MxNE, and, in particular, Champagne are not suited to determine the shape of the patch and would rather suggest the existence of several distinct point sources.

First, we consider

  two scenarios with two patches of medium distance: scenario InfFr & InfPa and scenario InfFr & MidTe. We use the same signals for the dipoles of both patches except for a 3 to 4 sample delay from one patch to another. The source imaging performances in terms of ROC, reconstructed sources, and DLE are shown in Fig. 7 and 8, and Table 3, respectively. The patch MidTe is partly located in a sulcus (cf. Fig. 2), leading to weaker surface signals than the patches InfFr and InfPa, which are mostly on a gyral convexity. This results in a deterioration of the performance of all source imaging algorithms, except for SVB-SCCD and Champagne, for the scenario InfFr & MidTe compared to the scenario InfFr & InfPa. More particularly, for the scenario InfFr & InfPa, all source imaging algorithms exhibit high dipole amplitudes for dipoles belonging to each of the true patches, whereas for the scenario InfFr & MidTe, the weak patch is less visible on the estimated source distributions of cLORETA, MCE, and MxNE, slightly better visible on the sLORETA solution, but completely missing for 4-ExSo-MUSIC. SVB-SCCD and STWV-DA both recover the patch MidTe, but with smaller amplitude in case of SVB-SCCD and smaller size for STWV-DA. Next, we study two three-patch scenarios: InfFr & InfPa & SupOcc and InfFr & MidTe & OccTe. We consider a delay of 3 to 4 samples between the signals of the first two patches and a delay of 5 to 6 samples between the signals of the first and third patch. For the resulting ROC curves, source reconstructions, and DLEs, see Fig. 9 and 10, and Table 3. The best results are achieved by SVB-SCCD, followed by STWV-DA and cLORETA. STWV-DA leads to good results for the scenario InfFr & InfPa & SupOcc, but underestimates the extent of the patch MidTe for the scenario InfFr & MidTe & OccTe. For this scenario, STWV-DA also identifies several spurious source regions of small size located between the patches MidTe and InfFr. for some realizations. The cLORETA solution exhibits dipoles with high amplitudes at all three patch positions, but makes it difficult to determine the patch shape. The same problem is exacerbated for MxNE, MCE, and Champagne, which additionally identify a number of scattered dipoles around the three foci of brain activity. This explains the reduced performance compared to SVB-SCCD. 4-ExSo-MUSIC yields similar results to STWV-DA for the scenario InfFr & InfPa & SupOcc, but overestimates the size of the patch InfFr while recovering a smaller area for the patch SupOcc. In addition, the estimated source region corresponding to patch InfPa is not at the correct position, leading to a higher DLE and worse ROC curve than STWV-DA. For scenario InfFr & MidTe & OccTe, 4-ExSo-MUSIC recovers only one of the two patches located in the temporal lobe, namely the patch OccTe. Finally, sLORETA leads to blurred source reconstructions and does not permit to discern the close patches InfPa and SupOcc or MidTe and OccTe.

Figure 3

 3 Figure 3 Illustration of the ROC curves for the different single patch scenarios.

Figure 4

 4 Figure 4 Illustration of the recovered patches for the different single patch scenarios. Triangles belonging to the original patches are marked in red, correctly identified triangles are dark red and erroneously identified triangles are yellow.

Figure 5 ROC

 5 Figure 5 ROC curves obtained for the different source imaging methods for different sizes of the patch SupFr: (top left) 10 dipoles, (top right) 200 dipoles, (bottom) 400 dipoles.

Figure 6

 6 Figure 6Illustration of the recovered source distributions for the different patch sizes.

Figure 7 ROC

 7 Figure 7 ROC curves obtained for the different source imaging methods for the scenario InfFr & InfPa (left) and the scenario InfFr & MidTe (right).

Figure 8

 8 Figure 8Illustration of the original patches and the recovered source distributions for the two considered two-patch scenarios.

Figure 9 ROC

 9 Figure 9 ROC curves obtained for the different source imaging methods for the scenario InfFr & InfPa & SupOcc (left) and the scenario InfFr & MidTe & OccTe (right).

Figure 10 Illustration

 10 Figure 10Illustration of the original patches and the recovered source distributions for the two considered three-patch scenarios.

Figure 1 :

 1 Figure 1: Illustration of a realistic head model with three compartments representing the brain, the skull, and the scalp and a source space that consists of a large number of dipoles (represented by black dots) located on the gray matter/white matter interface.

Figure 2 :

 2 Figure 2: Location of the 11 patches that are considered for the simulations in this paper.

Figure 4 :Figure 5 :Figure 8 :

 458 Figure 4: Illustration of the recovered patches for the different single patch scenarios. Triangles belonging to the original patches are marked in red, correctly identified triangles are dark red and erroneously identified triangles are yellow.

Table 1 :

 1 Performance of source imaging algorithms in terms of DLE (in mm) for the 8 different single patch scenarios. The DLE values are averaged over 50 realizations with different spike-like signals and varying background activity. The smallest obtained DLE value for each patch is marked in bold.

	patch	InfFr InfPa Cing SupOcc PreC BasTe MidTe Hipp
	sLORETA	3.61	2.41	3.28	2.05	4.07	2.26	6.28	3.11
	cLORETA	1.43	2.72	24.8	1.28	3.17	3.56	7.43	11.1
	SVB-SCCD	1.06 0.13 6.44	0.37	3.33	1.41	0.74	6.22
	MxNE	8.18	6.18	27.9	4.53	10.6	9.89	8.53	18.4
	MCE	3.99	3.19	22.5	3.07	4.86	6.45	5.20	14.7
	Champagne	4.60	4.03	2.95	2.47	2.95	2.29	4.37	4.09
	STWV-DA	0.44 0.95	1.70	0.57	1.84 2.29	1.85	1.86
	4-ExSo-MUSIC 0.44 0.96	1.70	0.58	1.84 2.39	1.83	1.83

Table 2 :

 2 Performance of source imaging algorithms in terms of DLE (in mm) depending on the size of the patch SupFr. The DLE values are averaged over 50 realizations with different spike-like signals and varying background activity. The smallest obtained DLE value for each patch is marked in bold.

	InfFr & InfPa InfFr & MidTe

Table 3 :

 3 Performance of source imaging algorithms in terms of DLE (in mm) for the considered scenarios with two and three patches. The DLE values are averaged over 50 realizations with different spike-like signals and varying background activity. The smallest obtained DLE value for each scenario is marked in bold.

In the online supplementary material, a third evaluation criterion, the Patch Center Distance (PCD), which is independent of the patch size, is considered, but as the findings are similar to those of the DLE and ROC, they are not reported in the main article.

Acknowledgements H. Becker was supported by Conseil Régional PACA and by CNRS France. The work of P. Comon was funded by the FP7 European Research Council Programme, DECODA project, under grant ERC-AdG-2013-320594. The work of R. Gribonval was funded by the FP7 European Research Council Programme, PLEASE project, under grant ERC-StG-2011-277906. Furthermore, we acknowledge the support of Programme ANR 2010 BLAN 0309 01 (project MULTIMODEL).

Conflicts of interest