a fundamental , profound and broad - ranging unsolved question is how cooperation among animals and humans has evolved . from the point of view of natural selection ,the question to be answered is why cooperation may be a better strategy for survival or reproduction than a more selfish behavior . for this reasonevolutionary game theory has been the mathematical framework that has provided the deepest insights into this issue .simple games such as the prisoner s dilemma , the snowdrift game or the stag - hunt game have been the subject of intense experimental and theoretical work along this line .one of the main achievements of this approach has been to show that the emergence of cooperation is sensitive to whether populations are well - mixed , such as in replicator dynamics evolution , or spatially structured .co - evolution of agents and networks and finite population effects are also relevant factors to take into account inasmuch as they may enhance or hinder cooperation . however, none of these approaches has considered the influence of different selection rates so far , surely because ever since darwin it has been acknowledged that natural selection acts at a very slow pace .nevertheless , recent experiments show that this may not always be the case : under certain circumstances ( e.g. strong predation or captive breeding ) evolution selects for a new trait in just a few generations .generally , research on evolutionary game theory is based on a population of individuals or agents that interact by playing a game . in the absence of spatial structure, it is posited that every agent plays the game against every other one , and then reproduction proceeds according to the payoffs earned during the game stage . for large populations , this amounts to saying that every player gains the payoff of the game averaged in the current distribution of strategies . in terms of time scales, such an evolution corresponds to a regime in which selection takes place at a much slower rate than the interaction between agents .however , these two time scales need not be different in general and , in fact , for many specific applications they can arguably be of the same order .our main aim in this work is to show that the pace at which selection acts on the population is crucial for the appearance and stability of cooperation . even in non - dilemma games such as harmony , where cooperation is the only possible rational outcome, defectors may be selected for if population renewal is very rapid .similar results hold true for several others games , thus pointing out the necessity to include a discussion on the rate of selection , compared to the rate of interaction , in any study about cooperation or any other situation modelled by evolutionary games .when selection acts at a much slower rate than interaction , a widely used tool to analyze frequency - dependent selection without mutation is replicator dynamics .replicator dynamics assumes a well - mixed population where all agents interact before selection , with a per capita growth rate of each strategy proportional to its fitness ( the payoff earned in a round of games between selection events ) .however , as stated in the previous section , we aim to understand the effects of different selection rates . to this end, we introduce the following new dynamics : there is a population with players . a pair of individuals is randomly selected for playing , earning each one an amount of fitness according to the rules of the payoff matrix of the game .this game act is repeated times , choosing a new random pair of players in each occasion .after every games , selection takes place .following , we have chosen moran dynamics to model selection in a finite population .one individual among the population of players is chosen for reproduction proportionally to its fitness , and its offspring replaces a randomly chosen individual . as the fitness of all playersis set to zero before the following round of games , the overall result is that all players have been replaced by one descendant , but the player selected for reproduction has had a reproductive advantage of doubling its offspring at the expense of the randomly selected player .it is worth noting that the population size is therefore constant along the evolution .the parameter controls the time scales of the model , i.e. reflects the relation between the rate of selection and the rate of interaction . for selectionis very fast and very few individuals interact between selection events .higher values of represent proportionally slower rates of selection .thus , when selection is very slow and population is effectively well - mixed .it only remains to specify the games we will be studying in this paper , namely binary games .these have been widely used in evolutionary game theory , because of their simplicity and amenability for both analytical treatment and computer simulation .a symmetrical binary game is completely defined by its payoff matrix ( or normal form , see e.g.  ) whose rows give the payoff obtained by each strategy when confronted to all others including itself . when interpreted as a model for the emergence of cooperation ,c and d denote , respectively , the strategies `` cooperate '' and `` defect '' .different kinds of games have been defined according to the relations between the four coefficients of the matrix .the dynamics we introduced in the previous section can be studied analytically and exactly in the following manner : let us denote by the number of cooperators present in the population .we calculate the probability of ending up in state ( i.e. , all players cooperate ) when starting in state . for and , we obtain an exact , analytical expression for . for arbitrary values of , such a closed form can not be found ; however , it is possible to carry out a combinatorial analysis of the possible combinations of rounds and evaluate , numerically but exactly , .see appendix [ maths ] for the detailed mathematics .  as our first and most striking example of the influence of the selection rate, we will start by considering the harmony game , determined by .the _ only _ nash equilibrium of this game is , as it is obvious from the payoffs : the best option for both players is to cooperate , which yields the maximum payoff for each one . when this game is framed in our dynamical model , fig .[ fig:1]a shows that the rationally expected outcome , namely that the final population consists entirely of cooperators , is not achieved for small and moderate values of the selection rate parameter . for the smallest values , only when starting from a population largely formed by cooperators there is some chance of reaching full cooperation ; most of the times , defectors will eventually prevail and invade the whole population .this counterintuitive result may arise even for values of comparable to the population size , by choosing suitable payoffs ( not shown ) .interestingly , the main result that defection is selected for small values of does not depend on the population size ; only details such as the shape of the curves ( cf .fig.[fig:1]b ) are modified by .[ figure [ fig:1 ] here ]  in the preceding paragraph we have chosen the harmony game to discuss the effect of the rate of selection , but this effect is very general and appears in many other games . to see this , consider the example of the stag - hunt game , with payoffs .this is the paradigmatic situation of a game with two nash equilibria in pure strategies , one pareto - dominant  , in which players maximize their payoffs , and the other risk - dominant  , in which players minimize the possible damage resulting from a defection of the partner . which of these equilibria is selected has been the subject of a long argument in the past , and rationales for both of them can be provided . as fig .[ fig:2 ] shows , results for small are completely different from those obtained for larger values .indeed , we see that for , all agents become defectors except for initial densities of cooperators close to 1 . however , for values of the resulting curve is quasi - symmetrical , reflecting a much more balanced competition between both strategies .[ figure [ fig:2 ] here ]  yet another example of the importance of the selection rate is provided by the snowdrift game , defined by the payoffs .also known as chicken or hawk - dove , it is a dilemma game not unrelated to , but different from , the prisoner s dilemma .[ fig:3 ] shows that for small values of defectors are selected for almost any initial fraction of cooperators .when increases , we observe an intermediate regime where both full cooperation and full defection have a nonzero probability , which , interestingly , is almost independent of the initial population . and , for large enough , full cooperation is almost always achieved .[ figure [ fig:3 ] here ]  finally , we show for completeness the results for the most ubiquitous game in studies about the evolution of cooperation : the prisoner s dilemma , with payoffs .as is well known , in this game the rational choice is to defect .the effect of a small number of games , shown in fig .[ fig:4 ] , is to bias the game even more towards defection . in any case, the initial density of cooperators must be very large for cooperation to have some chance of becoming the selected strategy , but for small values of this requirement is most severe .therefore , the parameter does not change the qualitative behavior of the prisoner s dilemma , alhtough once again low works against cooperation .[ figure [ fig:4 ] here ]  similar results are to be found in almost any binary game .indeed , it can be shown that d strategists are selected for when the payoffs satisfy in the extreme case , irrespective of , , or the population size ( cf .( [ 3b ] ) in appendix [ maths ] ) .larger values can not be analyzed in such a simple manner and , in particular , the corresponding results depend on all four parameters of the payoff matrix .the reason for this dramatic influence of the parameter resides in the resulting fitness distribution over the population . for large values of , most agents have played between selection events , and as a consequence most of the population has nonzero payoff , which in turn implies a nonzero probability of being selected for reproductive advantage . on the contrary , with small  distribution concentrates on the few players that have actually played , these ones being the only candidates for selection .this results in a completely different probabilistic scenario , with the consequences reflected in the results above .  as a specific example , consider the stag - hunt game shown in fig.[fig:2 ] , for the particular initial number of cooperators . with values of ,the payoffs are distributed to a large set of pairs of players , with approximate frequencies of  ,  and  or .given that the payoff matrix fulfills , both strategies collect a practically equal amount of fitness , thus with no reproductive advantage for any strategy and then .however , considering the case of , only a few pairs are selected to play . with pairs or ,the reproductive advantage is obviously the same , as only one strategy receives all the fitness . but pairs or will draw a strong advantage to defectors , given the relation , which in the end causes .results for other games can be understood in a similar way .let us now summarize our main findings . we have shown that selection rate plays a crucial role in determining the fate of cooperation , by studying how evolutionary dynamics in a moran setting depends on the number of times the game is played between selection events .we have seen that even in a game as simple as harmony , where cooperating is the only rational outcome , rapid selection leads to the success of defectors .we have observed similar behavior in other examples such as the snowdrift and the stag - hunt games . in binary games about cooperation situations ( including prisoner s dilemma ) , we have found that rapid selection rates generally lead to the promotion of defectors . in other contextsthe interpretation of the results would be different ( such as a choice of the risk - dominant coordination option in the stag - hunt game ) but the effect of the selection rate will undoubtedly be there .it is important to stress that the results are fully analytic , involve no approximation , and apply to both finite and infinite populations .although in this paper we have worked only in the framework of binary games , we believe that our main claim , namely that different time scales for interaction and selection can modify the outcome of evolution , is relevant to evolutionary games in general .consider , for instance , the ultimatum game , which is one of the most frequently used games in theoretical and experimental studies of cooperation .this game is much more complex than the previous binary games , as it asymmetrical , i.e. each individual of the pair of interacting players has a role , and there is a large number of strategies , not just two . in the ultimatum game , under conditions of anonymity ,two players are shown a sum of money .one of the players , the `` proposer '' , is asked to offer an amount of this sum to the other , the `` responder '' .the proposer can make only one offer , which the responder can accept or reject .if the offer is accepted , the money is shared accordingly ( the proposer receives the rest of the money ) ; if rejected , both players gain nothing .since the game is played only once ( no repeated interactions ) and anonymously ( no reputation gain ) , a self - interested responder will accept any amount of money offered whereas a self - interested proposer will offer the minimum possible amount which will be accepted .this is precisely the outcome predicted by the replicator dynamics in a well - mixed population : the final population will consist only of fully rational agents which offer the smallest possible amount and accept any amount . on the contrary , relaxing the assumption of a well - mixed population in favor of a dynamics that allows a faster selection rate changes this result absolutely . proceeding similarly to the binary games of the previous section , pairs of playersare randomly selected to interact , in series of rounds between reproduction - selection events .exhaustive simulations of different versions of the game with this kind of dynamics have shown that fair split becomes then the dominant strategy .an analytical study of this problem that confirms the simulation results is under way .  finally , the most important implication of our results is that , in studies about the emergence of cooperation , the rate of selection is an extremely influential parameter and very often leads to non - trivial , unexpected outcomes .of course , the scope of this result is not limited to cooperation among humans , as cooperative phenomena have been reported for many other species including bacteria .in fact , the research reported here stresses the importance of selection rates for evolutionary game theory , for all the situations it models ( not only cooperation ) and for evolutionary theory in general .in moran dynamics , at each time step an individual is chosen for reproduction proportional to its fitness , and one identical offspring is produced that replaces another randomly chosen individual . in a population of individuals where are c strategists ( cooperators ) and are d strategists ( defectors ) , we have a markov process with a tridiagonal transition matrix ( a birth - death process ) given by and , where is the fitness earned by cooperators ( ) or defectors ( ) after games , and denotes the average over realizations of the process .the dependence of the calculation on the selection rate , i.e. , on , enters only in these two quantities .the solution to this birth - death process is obtained in a standard manner .let us denote by the probability of ending up in state when starting off from state .then we have with boundary conditions , . the solution to this equationis given by when the ratio of the transition probabilities can be written as equation ( [ 6 ] ) has the closed form ,\ ] ] where the generalized binomial coefficient is defined as .for the extreme case there are two possible outcomes of the variable , namely , with probability ( that of pairing two cs ) , and , with probability ( that of pairing a c and a d ) .thus , we have .\ ] ] using a similar reasoning for we end up with which has the form ( [ 7 ] ) .notice that if then regardless the values of and .  in the opposite limit every player plays with every other an infinite number of times , so where denotes the payoff of strategists when every player plays with every other once .thus , in this case which has the form ( [ 7 ] ) only when .  for other values of , and can be computed through a combinatorial enumeration of all possible pairings , and the resulting exact expression ( too cumbersome to be included here ) can then be evaluated numerically .we acknowledge financial support from ministerio de educacin y ciencia of spain through grants bfm2003 - 07749-c05 - 01 , fis2004 - 01001 and nan2004 - 09087-c03 - 03 ( as ) , bfm2003 - 0180 ( jac ) , the thematic network fis2004 - 22008-e and the action fis2004 - 22783-e .support from the comunidad de madrid programme for research groups at universidad carlos iii and from the european science foundation through the cost action `` physics of risk '' is also acknowledged .probability of ending up with all cooperators starting from cooperators , for different values of , in the harmony game .a ) for the smallest values of , full cooperation is only reached if almost all agents are initially cooperators .values of of the order of 10 show a behavior much more favorable to cooperators . in this plot , the population size is .b ) taking a population of , we observe that the range of values of for which defectors are selected does not depend on the population size , only the shape of the curves does .parameter choices are : number of games between selection events , , as indicated in the plots ; payoffs for the harmony game , , , , .the dashed line corresponds to a probability to reach full cooperation equal to the initial fraction of cooperators and is shown for reference .same as fig .[ fig:1 ] , for the stag - hunt game . the probability of ending up with all cooperators when starting from cooperators , is very low if is small , and as increases it tends to a quasi - symmetric distribution around .parameter choices are : population , ; number of games between selection events , , as indicated in the plot ; payoffs for the stag - hunt game , , , , .same as fig .[ fig:1 ] , for the snowdrift game .the probability of ending up with all cooperators starting from cooperators is almost independent of , except for very small or very large values .small values lead once again to selection of defectors , whereas cooperators prevail more often as increases .parameter choices are : population , ; number of games between selection events , , as indicated in the plot ; payoffs for the snowdrift game , , , , .same as fig .[ fig:1 ] , for the prisoner s dilemma , but only the rightmost part of the axis is shown ( smallest values have a negligible chance to give rise to cooperation ) . in this game ,small values of lead to an even larger possibility of defection .parameter choices are : population , ; number of games between selection events , , as indicated in the plot ; payoffs for the prisoner s dilemma , , , , .

Abstract: how cooperation emerges in human societies is still a puzzle . evolutionary game theory has been the standard framework to address this issue . in most models , every individual plays with all others , and then reproduce and die according to what they earn . this amounts to assuming that selection takes place at a slow pace with respect to the interaction time scale . we show that , quite generally , if selection speeds up , the evolution outcome changes dramatically . thus , in games such as harmony , where cooperation is the only equilibrium and the only rational outcome , rapid selection leads to dominance of defectors . similar non trivial phenomena arise in other binary games and even in more complicated settings such as the ultimatum game . we conclude that the rate of selection is a key element to understand and model the emergence of cooperation , and one that has so far been overlooked .   [ [ keywords ] ] keywords + + + + + + + +  selection rate , cooperation , evolution , game theory  [ [ short - title ] ] short title + + + + + + + + + + +  selection rate and evolution of cooperation


network neutrality is an approach to providing network access without unfair discrimination among applications , content or traffic sources .discrimination occurs when there are two applications , services or content providers that require the same network resources , but one is offered better quality of service ( shorter delays , higher transmission capacity , ) than the other . how to define what is `` fair '' discrimination is still subject to controversy .effective performance of its statutorily mandated responsibilities '' .the fcc was deemed , then , unable to sanction discriminatory practices on internet s traffic carried out by american isps , and the underlying case on the `` fairness '' of their discriminatory practices was not even discussed . ] a preferential treatment of traffic is considered fair as long as the preference is left to the user . throughout the world by several public consultations for new legislations on the subject .the first one , proposed in the usa , was looking for the best means of preserving a free and open internet .the second one , carried out in france , asks for different points of view over net neutrality .a third one is intended to be presented by the eu during summer 2010 , looking for a balance on the parties concerned as users are entitled to access the services they want , while isps and cps should have the right incentives and opportunities to keep investing , competing and innovating .see . ]internet service providers ( isps ) may have interest in traffic discrimination either for technological or economic purposes .traffic congestion , especially due to high - volume peer - to - peer traffic , has been a central argument for isps against the enforcement of net neutrality principles .however , it seems many isps have blocked or throttled such traffic independently of congestion considerations .isps recently claimed that net neutrality acts as a disincentive for capacity expansion of their networks . in , the authors studied the validity of this argument and came to the conclusion that , under net neutrality , isps invest to reach a social optimal level , while they tend to under / over - invest when neutrality is dropped . in their setting ,isps stand as winners while content providers ( cps ) are left in a worse position , and users who pay the isps for preferential treatment are better off while other consumers have a significantly worse service .isps often justify charging cps by quantifying the large amount of network resources `` big '' content providers use . on the other hand, the content a cp offers contributes to the demand for internet access , and thus benefits the access providers .many references advocate the use of the shapley value as a fair way to share profits between the providers , see , .one of the main benefits of this approach is that it yields pareto optimality for all players , and requires in particular that cps , many of whom receive third - party income such as advertising revenue from consumers demand , help pay for the network access that makes this new income possible .  in this paper, we focus on violations of the neutrality principles defined in where broadband service providers  * charge consumers more than `` only once '' through usage - based pricing , and * charge content providers through side - payments .  within a simple game - theoretic model ,we examine how regulated side payments , in either direction , and demand - dependent advertising revenues affect equilibrium usage - based prices . we also address equilibria in stackelberg leader - follower dynamics .  the rest of the paper is organized as follows . in section [ basic - sec ] , we describe a basic model and derive nash equilibria for competitive and collaborative scenarios .we consider potentially non - neutral side - payments in section [ side - sec ] and add advertising revenues in section [ ad - sec ] , analyzing in each case how they impact equilibrium utilities .we study an isp offering multiple service classes in section [ diffserv - sec ] , and generalize our model in section [ multicp - sec ] to non - monopolistic content or access providers . in section [ stackelberg - sec ] , we consider leader - follower dynamics .we conclude in section [ conclusion ] and discuss future work .our model encompasses three actors :  * the internauts ( users ) , collectively , * a network access provider for the internauts , collectively called isp1 , and * a content provider and its isp , collectively called cp2 .the two providers play a game to settle on their ( usage - based ) prices .the internauts are modeled through their demand response .consumers are assumed willing to pay a usage - based fee ( which can be ] .  *if , nep conditions imply utility at the nep is therefore .\left[2 + \left(2 e^{-\delta } - 3\right ) \delta \gamma\right]\ ] ] in this setting , the value of is upper bounded by which is achieved when ( recall that is not a decision variable ) . * if , then and , yielding   hence , irrespective of consumers sensitivity to the price gap , the best solution for the coalition is to set - up usage - based pricing for content only , at price , while network access is subject only to flat - rate pricing ( ) .now consider splitting the demand - response coefficient into , and , that is : if then the interior equilibrium conditions yield : when the demand - response coefficients satisfy ( [ response - rel ] ) , we have an _equilibrium line_. vector field plots of suggest it is attractive ( see figure [ fig - split ] ) . in this particularsetting , providers can thus reach with usage - based pricing .    however , if , there exists a line of attraction , but with a non - null gradient on it driving players toward border equilibria .hence , the conclusion of subsection [ multiple - collab - subsec ] also holds in this more generalized setting .   when isp1 and cp2 compete , again there is no interior nep ( with all prices strictly positive ) .in fact , the condition implies and , so isp1 has to relax condition by setting ( only flat - rate pricing for the best - effort service ) .the solution to the two remaining nash equilibrium conditions is then : \cdot \pmax \\ p_h & = & \frac\gamma{2 \sqrt{9\gamma^2 + 2\gamma + 1 } - 3\gamma + 2 } \cdot \pmax\end{aligned}\ ] ] by defining and , we then have figure [ fig - ut-1 ] shows utilities at equilibrium ( as fractions of ) .we see that , in any case , cp2 has the advantage in this game : is always greater to , irrespective of consumers sensitivity to usage - based prices .  here , means users are so sensitive to any usage - based price that they will always choose the best - effort service ( which is subject to flat - rate pricing ) .users price sensitivity decreases as increases , the limit corresponding to the setting of section [ basic - sec ] with .now suppose there are multiple cps supplying the _ same _ type of content ( competing online encyclopedias ) , so users choose one cp over another based only on price .for the sake of simplicity , let us consider the case with two cps denoted by cp2 and cp3 .first , let us remark that if there is a significant difference between the prices of cp2 and of cp3 , since both provide the same type of content , all consumers are likely to shift to the cheapest provider , leading us back to our initial model with one isp and one cp .so the difference introduced by multiple cps may arise when . suppose that , initially , in this case , we assume customers are evenly shared between cp2 and cp3 , so that where is the demand - reponse to the usage - based prices ( for network access ) and ( for content ) .now , if cp reduces its price by some small , some of its opponent s consumers will change cp , but not all of them since a small price gap may not convince them to go .this behavior is known as _ customer stickiness , inertia or loyalty_. to model it we rewrite as where denotes the usage - based price of the other cp , and the `` stickiness '' function has the following properties : when cp reduces its price by , the first - order variation in its utility is given by . from ( [ sticky - ui ] ) and ( [ sticky - s3 ] ) , d(p_1 , \bp ) \bp.\ ] ] ( where was defined as . ) thus , taking consumers loyalty into consideration , the nash equilibrium condition for either cp becomes :    as a first , simple loyalty model , suppose that after cp reduces its price by , the fraction of users that remain with the other cp is inversely proportional to its price , the stickiness function is in this setting , equilibrium condition ( [ sticky - equ ] ) becomes , while equilibrium condition for isp1 is .thus , prices at the nep are and .demand at equilibrium is and the revenue of each provider is we see that , compared to ( [ ui - basic - comp ] ) , isp1 highly benefits from the competition between cps ( his revenue is about higher ). the situation would be symmetric with a single cp and two competing isps .one can consider another loyalty model where the fraction of users remaining with cp is proportional to the price _ , the stickiness function is here condition ( [ sticky - equ ] ) becomes , while equilibrium condition for isp1 is still .resolution of this system leads to and . at the nep ,demand is thus and utilities are we see with this second setting that the outcome of the price war between cps and isp1 significantly depends on the customer inertia model used .now focusing on the first stickiness model ( [ sm1 ] ) and update the model to take into account usage - based side payments .the revenues become for same - priced cps , solving ( with non - nul demand ) , we find that the equilibrium conditions are they are now quadratic in , thus , for the sake of readability , let us define   when ( side payments from the cps to the isp ) , resolution of this system for positive prices lead us to : then , demand at the nep is while revenues are what is interesting here is that both utilities are monotone in ( see figure [ u - eta-1 ] ) : decreases while increases with ( when , we fall back to the results ( [ sm1-u1 ] ) and ( [ sm1-ui ] ) of subsection [ s1-subsec ] ) .  . ]paradoxically enough , we see that increasing ( which means more usage - based side payments for isp1 ) is disadvantageous for isp1 but benefits the cps !this situation is very different from the one in section [ side - sec ] , where the isp was favored over the cp when was over a fixed threshold .when , cps receive usage - based side payments from isp1 ( ostensibly for royalties of copyrighted content ) .if , then is always negative and will tend to zero .this means that the best strategy for cps is to offer their content only for a flat rate , thus increasing demand and making all their usage - based profits on side payments.  otherwise , if , then condition ( [ sticky - equ ] ) has two solutions : there are therefore two equilibria :  * and : this is the case we studied in the setting , demand and revenues at the nep are unchanged .this equilibrium is `` stable '' in the sense that . ] and for : if cps move slightly their prices around , they are incented to move back . * and : in this case , demand at equilibrium is and revenues are given by however , this equilibrium is `` unstable '' in the sense that and : if cps shift their prices from , they are incented to shift even more , which will lead them either to the other equilibrium or to ( again , no usage - based pricing for content , but there may additional revenue from flat - rate fees as well ) . + . ]the isp is better off at this new nep ( see figure [ fig - u - eta-2 ] ) : regarless of the ( regulated ) value of , his revenue is always higher here ( and the cps revenues are always lower ) than at the other nep .this fact is consistent with the `` unstability '' we observed : if the cps happen to leave this equilibrium , they are not incented to come back.  a similar story follows if one considers multiple competing isps with one cp .also , taking advertising revenues into consideration will complicate the above computations and affect the location and `` stability '' of the neps .stackelberg equilibrium corresponds to asymmetric competition in which one competitor is the leader and the other a follower .actions are no longer taken independently : the leader takes action first , and then the follower reacts .though the dynamics of the games are different from the previous study , equations ( [ u1-complete ] ) and ( [ u2-complete ] ) still hold , with fixed and regulated . in the following ,we need to assume that so that neps are reachable with positive prices .if isp1 sets , then cp2 s optimal move is to set this expression yields and .anticipating cp2 s reaction in trying to optimize , the best move for isp1 is thus to set which yields therefore , when isp1 is the leader , at the nep demand is and utilities are : suppose now that cp2 is the leader and sets first .similarly , we find : these values yield the same cost and demand for the internauts at the nep , while providers utilities become :   therefore , in either case of leader - follower dynamics , the leader obtains twice the utility of the follower at the nep ( yet , his revenue is not better than in the collaborative case ) .using a simple model of linearly diminishing consumer demand as a function of usage - based price , we studied a game between a monopolistic isp and a cp under a variety of scenarios including consideration of : non - neutral two - sided transit pricing ( either cp2 participating in network costs or isp1 paying for copyright remuneration ) , advertising revenue , competition , cooperation and leadership .  in a basic model without side - payments and advertising revenues ,both providers achieve the same utility at equilibrium , and all actors are better off when they cooperate ( higher demand and providers utility ) .   * when , providers shift their prices to fall back to the demand of the competitive setting with no side - payments ; * when , the provider receiving side payments sets its usage - based price to zero to increase demand , while it is sure to be better off than his opponent .when advertising revenues to the cp come into play , they increase the utilities of _ both _ providers by reducing the overall usage - based price applied to the users .isp1 and cp2 still share the same utility at equilibrium , and the increase in revenue due to advertising is quadratic .  we considered in section [ diffserv - sec ] the implications of _ service differentiation _ from the isp . in our model , when isp1 and cp2 cooperate , the best solution for them is to set - up usage - based prices for content only and flat - rate pricing for network access .however , when providers do not cooperate , the isp optimally offers its best - effort service for a flat rate ( zero usage - based cost ) , resulting in more usage - based revenue for the cp .we considered in section [ multicp - sec ] a generalization of our model to non - monopolistic , competing cps . for a simple customer inertia model , we found that regulated side - payments had a significant impact on equilibrium revenues for the isp and the cps :  * when side payments go to the access provider , his utility at the nep diminishes , while * when they go to the content providers , the three - player system has two equilibria : an unstable one in favor of the isp , and a stable one .chiu , j.c.s .lui , v. misra , and d. rubenstein , interconnecting eyeballs to content : a shapley value perspective on isp peering and settlement " , in _ proc .intl workshop on economics of networked systems ( netecon ) _ , pp .61 - 66 , 2008 .chiu , j.c.s .lui , v. misra , and d. rubenstein , `` on cooperative settlement between content , transit and eyeball internet service providers '' , in _ proc .acm intl conference on emerging networking experiments and technologies ( conext ) _ , 2008 .

Abstract: hahn and wallsten wrote that network neutrality `` usually means that broadband service providers charge consumers only once for internet access , do not favor one content provider over another , and do not charge content providers for sending information over broadband lines to end users . '' in this paper we study the implications of non - neutral behaviors under a simple model of linear demand - response to _ usage - based _ prices . we take into account advertising revenues and consider both cooperative and non - cooperative scenarios . in particular , we model the impact of side - payments between service and content providers . we also consider the effect of _ service discrimination _ by access providers , as well as an extension of our model to non - monopolistic content providers .


starfinder is a code developed within the frame of the eso papao ( currie et al . , ) program for the reduction of adaptive optics data .it was originally designed ( diolaiti et al . ) to analyze ao images of very crowded fields , like the pueo frame of the galactic center shown in this paper , which contains about 1000 detectable stars in a field of view of 13 arcsec .this image is an example of an ao observation : the psf is nyquist - sampled and characterized by a complex shape , showing a sharp peak , one or more fragmented diffraction rings and an extended irregular halo .moreover , due to the small field of view , the imaged region is approximately isoplanatic and the psf may be considered space invariant . under the assumptions of isoplanatism and nyquist - sampling , starfinder models the observed stellar field as a superpositionof shifted scaled replicas of the psf lying on a smooth background due to faint undetected stars , possible faint diffuse objects and noise .the procedure derives first a psf digital template from the brightest isolated field stars ; then a catalogue of presumed objects is formed , searching for the relative intensity maxima in the image frame . in the following step the images of the suspected starsare analyzed in order of decreasing luminosity . in this phasea catalogue including the accepted objects is formed and a synthetic image of the observed field is constructed , placing an intensity - scaled psf template in the position of each identified star .each suspected object in the original list is accepted on the basis of its correlation coefficient with the psf template ; the relative astrometry and photometry of the source are determined by means of a fit , taking into account the contribution of the local non - uniform background and of the already detected stars , known from the synthetic field .as the analysis proceeds , fainter and fainter sources can be successfully analyzed , discriminating their central peaks from the secondary bumps of light in the distorted diffraction rings of the neighboring more luminous already identified stars : in this way the synthetic field becomes more and more similar to the observed image .residual unexplained features of the image may be further analyzed checking for indications of blended groups , at separations smaller than the psf fwhm .starfinder should be regarded as a tool for high precision astrometry and photometry of crowded fields acquired under the above assumptions of accurate psf knowledge , isoplanatism and correct sampling .an application of starfinder to hst undersampled data handled by dithering techniques ( fruchter et al . ) has been shown in aloisi et al .( ) . in this paperit can be seen that the results obtained by our method are comparable to those obtained by daophot ( stetson ) .as far as the psf is concerned , the accurate knowledge of its features outside the central peak is fundamental to perform a deep study of a stellar field , achieving accurate photometry of faint stars and avoiding either false detections or star loss ( esslinger & edmunds ) .if the psf template can not be extracted directly from the field , due to extreme crowding or lack of bright isolated stars , starfinder can still be applied using a psf estimated by means of other methods , as the reconstruction technique proposed by v et al .( ) for ao observations or the tiny tim simulation software for hst ( krist & hook ) .much more intriguing and difficult to solve is the case of a field with space variant psf , either due to anisoplanatic effects as in ao observations or to design and control in hst images . in general the analysis of an anisoplanatic fieldrequires the knowledge of the local psf .a method to reconstruct the off - axis psf in ao imaging has been proposed by fusco et al .( ) .starfinder , in its present version , can analyze frames with space invariant psf or sub- frames in which the isoplanicity condition is nearly satisfyied , as will be shown in section 4 ; the extension to the space variant case is in progress , with preliminary results presented by diolaiti et al .( , ) .the paper is organized as follows : the general features of the algorithm are described in sect . 2 ; sect .3 deals with more technical aspects and might be skipped on a first reading ; the method is validated on simulated and experimental data in sect . 4 ; details on the idl codeare presented in sect . 5 ; sect .6 includes our conclusions and future plans .the accuracy of the psf estimate is primary in starfinder , since the psf array is used as a template for all the stars in the image to be analyzed .the user selects the most suitable stars , which are background subtracted , cleaned from the most contaminating sources around , centered with sub - pixel accuracy , normalized and superposed with a median operation .the centering is performed by an iterative shift of the stellar image in order to cancel the sub - pixel offset of its centroid ( see christou & bonaccini ) .the median operation , which is performed pixel - by - pixel , is preferred to the mean because it is less sensitive to anomalous pixels ( outliers ) which might appear in one or more stellar images among the selected ones .the retrieved psf is post - processed in order to reject any residual spurious feature and to smooth the noise in the extended halo .the psf extraction procedure also reconstructs the core of saturated stars by replacing the corrupted pixels with the central part of the psf estimate .accurate positioning is achieved by means of a cross - correlation technique , while the scaling factor is determined with a least squares fit to the wings of the star to repair . for a detailed description of the procedure ,see sect .at first we build a list of objects , the presumed stars , which satisfy the condition where is the observed intensity , the background emission and a fair detection threshold .the presumed stars are analyzed one by one by decreasing intensity . to illustrate a generic step of the analysis, we consider the object in the list , after the examination of the first .a small sub - image of fixed size is extracted around the object ( fig .[ figuno ] , left ) .this sub - image may contain brighter stars formerly analyzed , fainter objects neglected in the current step and features of other stars lying outside the sub - image .the information on the brighter sources is recorded in a synthetic stellar field ( fig .[ figuno ] , right ) , defined as the sum of two terms : a superposition of psf replicas , one for each star detected up to this point , and an estimate of the background , assumed to be non uniform in general.  the local contribution due to the brighter stars and the background , derived from the synthetic field , is subtracted from the sub - image .if a statistically significant residual remains , it is compared to the central core of the psf by means of a correlation check .if the correlation coefficient is higher than a pre - fixed threshold then the object of interest is rated similar to the psf and accepted .the accurate determination of its position and relative flux is attained by means of a local fit , in which the observed sub - image is approximated with the multi - component model described in sect .the actual size of the fitting region is comparable to the diameter of the first diffraction ring of the psf .this choice ensures that the information represented by the shape of a high - strehl psf is included in the fitting process to achieve better accuracy and prevents the growth of the number of sources to be fitted together . for the central object of our examplea single component fit is performed and the contribution due to the brighter stars is considered as a fixed additive term .a multi - component fit is performed when the star is in a very compact group , at separations comparable to the psf fwhm .if the fit is acceptable , the parameters of the new detected star are saved and those of the already known sources , which have been possibly re - fitted , are upgraded .the new star and an upgrade of the re - fitted sources are added to the synthetic field .this analysis is performed for each object in the initial list ( a flow - chart illustrating the operations of starfinder is in figure [ figdue ] ) . to achieve a better astrometric and photometric accuracy , at the end of the examination of all the objects ,the detected stars are fitted again , this time considering all the known sources .this step may be iterated a pre - fixed number of times or until a convergence condition is met .  at the end of the analysis ,it is possible to stop the algorithm or instead perform a new search for lost objects removing the detected stars and using an upgraded background estimate .it should be stressed that this image subtraction is just a tool to highlight significant residuals .any further analysis is performed on the original frame , in order to take into account the effects arising from the superposition of the psfs of neighboring sources .generally , after 2 - 3 iterations of the main loop , the number of detected stars approaches a stable value .a binary star with different separation values ( fig . [ figtre ] ) has been simulated to show how the code works with crowded sources . with a separation of 2 psffwhm the two components are well separated and the code analyzes them with the standard procedure described in the previous sub - section . in the other cases ( separation from 1 to 0.5 psf fwhm ) the secondary component is not detected as a separate relative maximum and it is lost . however ,if the separation is not as small as 0.5 fwhm , a further iteration of the main loop enables the algorithm to detect the fainter component by subtracting the brighter one .this strategy forces the two stars to pass the correlation test , the principal component as a presumed single object and the secondary in a subsequent iteration of the loop . in a waythe iteration of the main loop is a de - blending strategy , because it enables the algorithm to detect stars whose intensity peak is not directly visible in the observed data .this strategy fails when :  * the binary is very noisy and the two components have similar flux . in a similar situationthe principal component may not pass the correlation test , preventing also the detection of the secondary in a further iteration .these noisy blends may be recovered at the end of the overall analysis .* the two components are almost equally bright and have a separation close to the lower limit ( 1/2 psf fwhm ) .the residual corresponding to the secondary after subtracting the principal component may have a distorted shape and might not pass the correlation check .the latter case may be handled by a method based on a thresholding technique .the object is cut at a prefixed level , about 20% below the central peak , and transformed to a binary array , setting to 1 all the pixel above the threshold and to 0 the pixels below .if the area of the pixels with value 1 is more extended than the psf , the object is considered a blend and the secondary star may be detected by subtracting the brighter one ; then a two - components fit allows accurate astrometry and photometry of the two sources .this strategy can be iteratively applied to multiple blends .it should be stressed that the area measurement is not reliable when the value of the cutting threshold is comparable to the noise level : for this reason the de - blending procedure is applied only to objects with a suitable signal - to - noise ratio .moreover the area measurement is reliable when the data are adequately sampled .this de - blending procedure is applied at the end of the last iteration of the main loop , when all the resolved sources have been detected : in this way the probability that a single object may appear artificially blurred because of the contamination of still unknown sources is largely reduced .  in a normal case , like the simulated field of sect .4.1 , two or three iterations of the main loop find almost all ( % ) the stars that starfinder may detect .the de - blending procedure described above adds % more stars , without additional false detection .the number of lost objects belonging to the first category described above is negligible ( % ) .normally we perform two or three iterations of the main loop and apply the de - blending strategy only in very crowded fields .starfinder includes an automatic procedure to repair known bad pixels , which are replaced with the median of the good data points in a suitable neighborhood .the use of this procedure is particularly recommended in the extraction of the psf .indeed the stellar images , which will be combined to form the psf , are centered with sub - pixel accuracy by means of an interpolation technique and the presence of bad pixels may produce unpredictable interpolation errors . even if replaced , the bad pixels are excluded from any further computation concerning the correlation coefficient and the fitting process .a reliable estimate of the background is necessary to define the detection condition and to compute the correlation coefficient of the presumed stars with the psf .  a straightforward technique to estimate the background is to smooth the image by median filtering , replacing each pixel with the median computed over a suitable neighborhood , of size comparable to the characteristic width of a stellar image .this method tends to over - estimate the background underneath strong peaks .a more accurate approximation , even in crowded fields , is described in bertin & arnouts ( ) .the image is partitioned in sub - regions arranged in a regular grid and a local estimate is calculated for each sub - patch by means of an idl implementation of the daophot sky routine ( stetson ) , due to landsman ( ) .this array of sky measurements is smoothed by median filtering and interpolated onto the same grid of the input image .it should be stressed that the background computation is unavoidably affected by the presence of bright sources . in generala more accurate estimate can be obtained after the analysis of the stellar field , when the most contaminating sources are known and can be subtracted .the estimate of the noise is useful to define the detection threshold and to compute the formal errors on the retrieved astrometry and photometry .the overall effect of photon and instrumental noise can be computed if the required parameters ( detector gain , read - out - noise , dark current , etc . ) are known .otherwise an estimate of the mean background noise can still be obtained by means of histogram fitting techniques ( almoznino et al . , bijaoui ) . assuming that the intensity of the sky radiation is distributed normally around a typical value, the histogram of the observed intensity levels should be quite similar to a gaussian distribution , whose mode and standard deviation represent respectively the sky value and the associated noise . actually the contamination due to stellar sources produces a high - intensity tail and an artificial broadening of the histogram , which prevents an accurate estimate of the background noise .this problem can be partially overcome by removing the signal from the image , leaving only the pixel - to - pixel variations associated to pure noise : a reasonable estimate of the signal to subtract for this purpose can be obtained by smoothing the data with a median filtering technique .after this operation the histogram is symmetric around its mode and the background noise standard deviation can be estimated by means of a gaussian fit to the histogram itself .false detections , associated to noise spikes or residual psf features of bright stars , are recognized on the basis of their low correlation coefficient with the psf , which represents a template for each true star in the image .the correlation coefficient ( gonzalez & woods ) is defined as \left [  p\left(x - a , y - b\right ) -\bar{p}\right ] } { \sqrt{\sum_{x , y } \left [ i\left (  x , y\right ) -\bar{\imath}\right]^{2}}\sqrt{\sum_{x , y } \left [ p\left ( x - a ,y- b\right ) - \bar{p}\right ] ^{2}}}\end{aligned}\ ] ] where and are the object and the psf respectively , and are the corresponding mean values .maximizing the correlation coefficient as a function of the offset yields an objective measure of similarity . after maximizing for integral offsets , it is possible to repeat the procedure for sub - pixel shifts , improving the positioning accuracy .the correlation coefficient is computed on the core of the psf : typically the central spike of the diffraction pattern is considered , out to the first dark ring .a fair correlation threshold must be fixed in order to discriminate and reject unlikely detections , without losing faint stars contaminated by the background noise ; a value of 0.7 or 0.8 is acceptable in most cases .the correlation coefficient represents also an effective tool to select the stars with the highest photometric reliability , since generally a very high correlation value is associated to resolved single sources .saturated stars provide precious information on the psf halo , so it may be useful to include them , appropriately reconstructed , in the psf extraction process .in addition , the repaired saturated stars can be recognized by the star detection algorithm and their contribution be taken into account during the analysis of near and fainter sources .the core of a saturated star is replaced with a shifted scaled replica of a preliminary estimate of the psf .the repaired star is defined as where is the upper linearity threshold of the detector .the position of the center is estimated by means of the maximization of the correlation coefficient , which is not sensitive to the intensity levels .the scaling factor is calculated with a least squares fit to the wings of the saturated star . of coursethe saturated pixels are excluded from all the computations .the background is temporarily subtracted in order to prevent affecting the repair process .this procedure has been applied to the brightest star ( irs 7 ) of the galactic center image shown in sect .this source is not saturated in the original data , but it has been artificially corrupted by an upper cut at half maximum . the repair procedure is able to reconstruct the original peak with an error of % , imposing a positioning accuracy of 1/2 pixel .   a sub - image centered on the star of interest is extracted and approximated with the model where is the fixed contribution of known stars outside the sub- image support , is the number of point sources within the sub - image , , , are the position and flux of the source , is the psf and , , are the coefficients of a slanting plane representing the local background .it should be stressed that the retrieved astrometry and photometry are referred to the absolute centering and normalization of the psf array .the optimization of the parameters is performed by minimizing the least squares error between the data and the model .if the noise is known , it is possible to weigh the data by their inverse standard deviation and obtain a statistically optimal fit ( beck & arnold ) . in this case ,formal error estimates on the parameters can be obtained ( bevington & robinson ) . the optimization is performed by means of an iterative newton - gauss technique with linearized hessian ( beck & arnold , luenberger ) . the moore - penrose generalized inverse concept ( rust & burrus , bendinelli et al . , lorenzutta ) is applied to invert the hessian matrix at every step .the iterations are stopped when the parameters approach a stable value .the major difficulty of a newton - like method is represented by the computation of the model derivatives with respect to the parameters , some of which ( stellar positions ) yield non - linear dependence . for this purpose the fourier shift theorem is applied : ^{-i2\pi ( ux_{n}+vy_{n})/n}\right]\end{aligned}\ ] ] where represents the discrete fourier transform operation , is the sub - image size and , are spatial frequencies . the derivative with respect to is then ^{-i2\pi  \left(ux_{n}+vy_{n}\right )/n}\left(-i \frac{2\pi u}{n } \right)\right ] =  \nonumber \\  & & ft^{-1}\left [ \left ( -i\frac{2\pi u}{n}\right ) ft\left [ p\left(x - x_{n},y- y_{n}\right ) \right ] \right ] \end{aligned}\ ] ] and requires in practice an interpolation of the psf to compute . in principlethis interpolation - based method can only be applied to nyquist - sampled images and this is currently the main limit of the algorithm .a similar technique has been described by v & rigaut ( ) .a band - limited function in one dimension is nyquist - sampled if the step size fulfills the condition where is the so - called cut - off frequency of the spectrum . in this caseit is possible to reconstruct the original continuous function from a set of equally - spaced samples by means of the so - called sinc interpolation ( mariotti ) .for a nearly diffraction - limited psf , the critical sampling condition is generally stated by saying that the its fwhm must contain at least two pixels .many accurate and efficient interpolation schemes exist to perform the psf shift required by the fitting procedure .probably the most simple is a straightforward application of the fourier shift theorem , but one may also use bicubic splines or the fast sinc interpolation algorithm described in yaroslavksy ( ) . in starfinderwe have applied the cubic convolution interpolation method ( park & schowengerdt ) , which approaches very closely the optimal sinc interpolation for nyquist - sampled data ; this algorithm is implemented in the idl function interpolate .  even though the interpolation of the psf array is allowed only on nyquist - sampled data, the cubic convolution technique seems to produce acceptable results even with marginally under - sampled images .tests performed on airy diffraction patterns , with a sampling step twice as large as the critical sampling step size , indicate that the interpolation - induced oscillations amount to a few percent of the image peak , as opposed to 10 - 20% of other interpolation techniques like fourier shift or bicubic splines .the code was first applied to a simulated image ( fig . [ figquattro ] ) including 1000 stars , placed randomly in the frame and with a given magnitude distribution ( fig .[ figsei ] ) .each star is a scaled copy of a long exposure high - strehl psf , obtained with the adonis ao system at the eso 3.6 m telescope .the image is 368 368 pixels large ( ) and has a stellar density of 6 stars arcsec .photon , readout noise and a background nebulosity , normalized to the same flux of the stellar sources , were added to the image .the faintest stars have a peak signal - to - noise ratio of 5 .we performed a standard reduction of the artificial image using the `` default '' parameters of the method .the psf was estimated by superposing the images of the four brightest stars in the field .the retrieved psf is very similar to the true one ( fig .[ figcinque ] ) .figure [ figsei ] shows the good agreement between the true and the observed luminosity function .the sole statistically significant discrepancy is in the bin from magnitude 7 to 8 , where % of the stars were lost ; the other small differences are due to photometric errors which shift some objects to a neighboring magnitude interval .the lost stars are % of the total number of sources and are generally faint : about 90% have magnitude between 7 and 8 , the rest is in the bin between magnitude 6 and 7 .the only bright lost star has magnitude .5 and is the secondary component of a very close binary , with a separation of just 1/2 pixel .roughly 70% of the lost stars are located at a distance 1 psf fwhm from the nearest object , % are on the first diffraction ring of a brighter source and only % are isolated .it should be stressed however that % of the lost stars can be recognized by visual inspection as faint objects in the halo of the four brightest stars in the field , independently of their separation from the nearest source .it is apparent that the blending effect and the contamination by the halo of very bright stars do account for the lost stars .note that the number of false detections in this simulated field is negligible ( 1 case out of 1000 ) .the plots in figures [ figsette ] , [ figotto ] show the astrometric and photometric accuracy of starfinder . about 80% of the detected stars have both astrometric error smaller than 0.1 psf fwhm and photometric error smaller than 0.1 magnitudes .the stars with less accurate astrometry or photometry are generally faint : % belong to the magnitude interval from 7 to 8 , % to the bin between magnitude 6 and 7 and only % are distributed in lower bins of the luminosity function .these stars have , in % of the cases , a lost source in their immediate neighborhood within the first diffraction ring of the psf .the others are faint stars dispersed in the halo of the brightest sources or in the noisy background nebulosity .  after discussing the performance of the code with a standard analysis, it is interesting to examine how the results are affected by the main parameters of the method . applying the de - blending strategy we detected + 10% more binaries in the separation range between 1/2 and 1 psf fwhm , even though the overall detection gain is less than 1% referred to the total number of sources . with a higher number of iterations of the main loop we detected + 30% more binaries in the range from 1 to 2 psf fwhm . decreasing the detection threshold from 3 to 1 times the noise standard deviation, we found + 25% more binaries in the range between 1/2 and 1 psf fwhm , but with 10 faint ( mag ) false detections instead of 1 . increasing the threshold on the correlation coefficient , from 0.7 to 0.8 , we reported no false detection , but the number of lost stars increased by about 60% ; the additional lost sources were generally fainter than magnitude 7 , but not necessarily in crowded groups . lowering the correlation threshold to 0.6 we detected more faint isolated stars and binaries , at separations between 1 and 2 psf fwhm , but with a higher probability of false detections ( 2 instead of 1 ) .finally the astrometric and photometric accuracy approaches a stable level after a few ( ) re - fitting iterations .the code was run on a 15 min exposure time k band ( 2.2 m ) image of the galactic center ( fig .[ fignove ] ) , taken with the pueo ao system on the 3.6 m cfh telescope ( rigaut et al .the strehl ratio in the image is % .the psf fwhm is , with a sampling of /pixel .the adaptive optics guide star was a =14.5 star ( called star 2 in biretta et al . ) located about ( to the upper left ) from the center of the image , out of the field of view of the figure . there is therefore a slightly elongation of the psf towards the direction of the guide star .however , due to the fact that the isoplanatic patch was much larger than the shown in the figure , a space - invarient psf fits the data very well , as we will show .a standard analysis was performed , analogous to the one described in sect .4.1 for the synthetic stellar field .about 1000 stars were detected , with a correlation coefficient of at least 0.7 ; the reconstructed image is shown in figure [ fignovea ] .we evaluated the accuracy of the algorithm by means of an experiment with synthetic stars .we created 10 frames adding to the original image 10% of synthetic stars at random positions for each magnitude bin of the estimated luminosity function .the 10 frames were analyzed separately .as in the simulated case , a distance tolerance of 1 psf fwhm was adopted to find coincidences between the detected stars and their true counterparts .the lists of detected artificial stars were merged together and the astrometric and photometric errors were computed and plotted as a function of the true magnitude ( figs .[ figdieci ] , [ figundici ] ) .the plots show no apparent photometric bias and high astrometric and photometric accuracy : the stars brighter than magnitude 5 , for instance , have a mean astrometric error mas and a mean absolute photometric error mag .it should be stressed however that the artificial sources are contaminated by the background noise present in the observed data and by the photon noise due to neighboring stars , but no additional noise was added .figure [ figdodici ] shows a comparison between the mean luminosity function retrieved in the 10 experiments and the truth . assuming an expected error for each bin equal to the square root of the corresponding number of counts , according to the poisson statistic, the only significant differences occur for magnitudes fainter than 9 .it is also interesting to consider the magnitude distribution of the false detection cases ( dashed - dotted line in figure [ figdodici ] ) , i.e. the detected stars for which we found no counterpart in the original list , within a distance of 1 psf fwhm .the mean percentage of false detections in the 10 experiments is 2% of the total number of stars .the false detections are almost always very faint ( mag ) ; their number is comparable to the square root of the total counts only in the last magnitude bin , for magnitudes fainter than 10 .the percentage of false detections reported in these experiments seems to confirm the analysis performed by visual inspection on the stars detected in the original frame .the starfinder code has been provided with a collection of auxiliary routines for data visualization and basic image processing , in order to allow the user to analyze a stellar field , produce an output list of objects and compare different lists , e.g. referred to different observations of the same target .the input image is supposed to be just calibrated .  the code is entirely written in the idl language and has been tested on windows and unix platforms supporting idl v. 5.0 or later . a widget - based graphical user interface has been created .the main widget appearing on the computer screen is nothing more than an interface to call secondary widget - based applications , in order to perform various operations on the image .the basic documentation about the code can be found in the on - line help pages .idl users might wish to run interactively the starfinder routines , without the widget facilities : complete documentation on each module is available for this purpose .the elaboration of real and simulated data seems to prove the effectiveness of starfinder in analyzing crowded isoplanatic stellar fields characterized by high strehl ratio psfs and correct sampling , reaching in this case the full utilization of the data information content .the code can be applied also to low strehl data with results comparable to those attainable by other methods .moreover we are evaluating its performance on undersampled images .starfinder is also reasonably fast : the analysis of the galactic center image ( pixels , stars ) requires between 5 and 10 minutes on a normal pc ( pentium pro - 64 mb ram - 350mhz ) .the graphic interface makes it accessible to users unfamiliar with idl .  in the near futurewe plan to provide the code with tools to handle images with spatially variable psf and new elements which might be helpful in the analysis of a stellar field .an interesting application , still in progress , aims at cleaning mixed fields from the contamination of foreground stellar images , leaving out true or suspected diffuse sources .franois rigaut is acknowledged for kindly providing the pueo image of the galactic center and for supporting the initial development of this method .the eso la silla 3.6 m telescope team is acknowledged for the support during the technical observations with adonis , from which we have retrieved real psfs used to simulate stellar fields .this work was partially supported by the eso instrumentation division adaptive optics dept . and by the italian ministry for university and research ( murst ) under grant cofin 98 - 02 - 32 .almoznino e. , loinger f. , brosch n. , 1993 , mnras 265 , 641 aloisi a. , clampin m. , diolaiti e. , greggio l. , leitherer c. , nota a. , origlia l. , parmeggiani g. , tosi m. , 2000,(in preparation ) beck j.v . ,arnold k.j . , 1977 ,parameter estimation in engineering and science , john wiley & sons bendinelli o. , parmeggiani g. , piccioni a. , zavatti f. , 1987 , aj 94 , 1095 bertin e. , arnouts s. , 1996 , a&as 117 , 393 bevington p.r . , robinson d.k . , 1994 , data reduction and error analysis for the physical sciences , mcgraw - 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Abstract: we describe a new code for the deep analysis of stellar fields , designed for adaptive optics ( ao ) nyquist - sampled images with high and low strehl ratio . the point spread function ( psf ) is extracted directly from the image frame , to take into account the actual structure of the instrumental response and the atmospheric effects . the code is written in idl language and organized in the form of a self - contained widget - based application , provided with a series of tools for data visualization and analysis . a description of the method and some applications to ao data are presented .


being able to distinguish between regular and chaotic dynamics in a deterministic system is an important question with applications ranging from cardiac arrhythmias to the stability of our solar system .much progress has been made in developing tests for chaos .recently we have introduced a binary test for chaos , the test , designed for the analysis of deterministic dynamical systems .the test distinguishes between regular and chaotic dynamics for a deterministic system .the nature of the dynamical system is irrelevant for the implementation of the test ; it is applicable to data generated from maps , ordinary differential equations and partial differential equations .the test has been applied to noisy numerical data  , experimental data  , quasiperiodically forced systems and strange nonchaotic attractors  , hamiltonian systems  , nonsmooth systems  and fluid dynamics  .  the usual test of whether a deterministic dynamical system is chaotic or nonchaotic involves the calculation of the maximal lyapunov exponent  .a positive maximal lyapunov exponent indicates chaos : if , then nearby trajectories separate exponentially and if , then nearby trajectories remain in a close neighbourhood of each other .this approach has been widely used for dynamical systems whose equations are known . if the equations are not known or one wishes to examine experimental data , then may be estimated using the phase space reconstruction method of takens , by approximating the linearisation of the evolution operator , or by the `` direct method ''  .in contrast our test does not depend on phase space reconstruction but rather works directly with the time series given .the main advantages of our test are ( i ) it is binary ( minimizing issues of distinguishing small positive numbers from zero ) , ( ii ) the nature of the vector field as well as its dimensionality does not pose practical limitations , and ( iii ) it does not suffer from the difficulties associated with phase space reconstruction .  in this paper , we describe in detail how to implement the test for chaos . in addition , we carry out modifications to the test that greatly improve the previous versions in  .  throughout the paper , we use the logistic map to illustrate our claims , with the exception of section [ sec - cont ] where we use the lorenz attractor as an example of a continuous time system . the reader can verify that our results apply equally well to other systems , including those considered in our previous papers  .   we briefly review how the test is implemented . given an observation for we perform the following sequence of steps :  1 .  for , we compute the translation variables for . typical plots of and for regular and chaotic dynamics are given in fig .[ fig - pq ] .+  versus for the logistic map . left : regular dynamics at ; right : chaotic dynamics at .we used data points ., title="fig : " ]  versus for the logistic map . left : regular dynamics at ; right : chaotic dynamics at .we used data points ., title="fig : " ] 2 .the diffusive ( or non - diffusive ) behaviour of and can be investigated by analyzing the mean square displacement .the theory behind our test assures that if the dynamics is regular then the mean square displacement is a bounded function in time , whereas if the dynamics is chaotic then the mean square displacement scales linearly with time . in section [ sec - msq ]we look at expressions for the mean square displacement and describe how one may use analytical expressions derived in to conveniently modify the expression for the mean square displacement .we then compute the asymptotic growth rate of the mean square displacement .methods for the most effective estimation of this are discussed in section [ sec - kc ] .steps 13 are performed for values of chosen randomly in the interval . in practice, is sufficient .the choice of is discussed further in section [ sec - c ] .we then compute the median of these values of to compute the final result .our test states that a value of indicates regular dynamics , and indicates chaotic dynamics .  in this paper, we explore practical issues arising in the implementation of the above algorithm .various issues associated with steps 24 are discussed in respectively . in section [ sec - confidence ]we examine finite data size effects . in particularwe look at weak chaos . in section [ sec - cont ]we consider continuous time systems where oversampled data can lead to small values of despite an underlying chaotic dynamics . in section [ sec - noise ]we investigate the issue of measurement noise .[ rmk - first ] in the first version of our test , introduced in , we defined and by iterating the extended system the current version of the test corresponds to the case .as shown in , the test with is less sensitive to measurement noise .[ rmk - valid ] it can be rigorously shown that ( i ) and are bounded if the underlying dynamics is regular , i.e. periodic or quasiperiodic and ( ii ) and behave asymptotically like brownian motion for large classes of chaotic dynamical systems .in we used results of  to prove this for the case in remark [ rmk - first ] . in the case these results are not applicable ; nevertheless in we cover the case under even weaker assumptions on the underlying dynamics .[ rmk - third ] in it was shown that the results are valid for almost all observables .of course , the choice of the observable influences the rate of convergence ( but not the limiting value or ) . from a practical point of view we found that changing the observable does not greatly alter the computed value of .for a given time series with , we compute the mean square displacement of the translation variables and defined in  for several values of . the mean square displacement is defined as ^ 2\ , + \ , [ q_c(j+n)-q_c(j)]^2 \ ; .\label{e.msq}\end{aligned}\ ] ] note that this definition requires .in we calculated the mean square displacement using directly the definition ( [ e.msq ] ) .the limit is assured by calculating only for where . in practicewe find that yields good results .the test for chaos is based on the growth rate of as a function of . in the following , we use analytical expressions derived in to formulate a modified mean square displacement which exhibits the same asymptotic growth as but with better convergence properties .  under mild assumptions on the underlying dynamical system , described in remark [ rmk - rho ] below , for each , where as uniformly in and the expectation is given by   the form ( [ e - mcn ] ) suggests an improvement for the test : we can subtract the explicit term from the mean square displacement and introduce note that the asymptotic growth rates of and are the same . +   for the logistic map with corresponding to chaotic dynamics .the oscillating ( green ) curve is the original mean square displacement as defined in ( [ e.msq ] ) ; the straighter ( red ) curve is the modified mean square displacement as defined in ( [ e.msqnew ] ) .we used data points and computed and for and . ]  in fig .[ fig - msq ] we show the two mean square displacements and for the logistic map with ( which corresponds to chaotic dynamics ) and an arbitrary value of .evidently , the subtraction of the oscillatory term regularizes the linear behaviour of .this allows a much better determination of the asymptotic growth rate .[ rmk - rho ] the autocorrelation function for the observation is given by provided the autocorrelations are absolutely summable ( that is , ) then equation ( [ e - mcn ] ) is valid , and moreover , the error term decays uniformly in ( see for example  ) .it is for this reason that the test based on greatly outperforms the test based on .furthermore , the absolute summability condition guarantees  that for all .this result follows from the birkhoff ergodic theorem , the wiener - khintchine theorem , and standard calculations .in particular , the slope of the mean square displacement is identified with the power spectrum .  for nonmixing systems, the error term no longer decays to zero and there are further oscillatory terms in addition to .nevertheless , the identification  remains valid for nonmixing systems under very weak conditions  .more importantly from the point of view of the test for chaos , working with remains highly advantageous even for nonmixing systems .this is illustrated for the logistic map in fig .[ fig - test ] later in this paper ,having calculated the modified mean square displacement for , the next step is to estimate the asymptotic growth rate .we have tried out two different methods : a _ regression _ method and a _ correlation _ method , described in subsections [ sec - regress ] and [ sec - corr ] below .the regression method consists of linear regression for the log - log plot of the mean square displacement . in we used the original mean square displacement , so the asymptotic growth rate is given by the definition , is determined by fitting a straight line to the graph of versus through minimizing the absolute deviation .  in section [ sec - msq ] , we demonstrated the superiority of the modified mean square displacement when compared to , so it is natural to apply the regression method to .whereas is strictly positive , may be negative due to the subtraction of the oscillatory term .hence , we set and obtain the asymptotic growth rate again , can be determined numerically by regression ( minimizing the absolute deviation ) for the graph of versus .   as a function of for the logistic map at .we used and calculated the mean square displacement up to . ][ rmk - regress ] minimizing the absolute deviation is preferable when compared to the usual least square method as the latter assigns a higher weight to outliers . since the linear behaviour of the mean square displacementis only given asymptotically , one typically encounters outliers for small values of .we find that it is usually sufficient to use the absolute deviation for estimating , and that it is not necessary to employ more complicated higher - order regression methods such as the method by yohai .the finite value of the -term in the definition of in ( [ e - mcn ] ) leads to a distortion for small values of . in such situations, one typically observes a flattening of the slope of or as illustrated in fig .[ fig - logm ] .it is those values for small of ( or ) which would be overestimated in a least square fit .we now present an alternative method for determining from the mean square displacement .( the method is described in terms of , but we could use instead . )  form the vectors and . given vectors , of length , we define covariance and variance in the usual way : now define the correlation coefficient \;.\end{aligned}\ ] ] this quantity measures the strength of the correlation of with linear growth .again , it can be shown rigorously  that under weak conditions on the underlying dynamics ( as described in remark [ rmk - valid ] ) we obtain for regular dynamics and for chaotic dynamics .   versus for the logistic map with increased in increments of .we used data points .the darker ( red ) lines are obtained by using the original definition of the mean square displacement in ( [ e.msq ] ) .the lighter ( green ) lines are obtained by using the modified mean square displacement in ( [ e.msqnew ] ) .the resulting values of are shown for the regression method ( left ) and the correlation method ( right ) .the horizontal lines ( blue and magenta ) indicate the cases and .we used values of ., title="fig : " ]  versus for the logistic map with increased in increments of .we used data points .the darker ( red ) lines are obtained by using the original definition of the mean square displacement in ( [ e.msq ] ) .the lighter ( green ) lines are obtained by using the modified mean square displacement in ( [ e.msqnew ] ) .the resulting values of are shown for the regression method ( left ) and the correlation method ( right ) .the horizontal lines ( blue and magenta ) indicate the cases and . we used values of ., title="fig : " ]  in practical terms , the correlation method greatly outperforms the regression method .this is evident from fig .[ fig - test ] which compares the regression method and the correlation method ( using both and ) for the logistic map .in fig . [ fig - cscan ] we show the asymptotic growth rate as a function of for regular and chaotic dynamics . in the case of periodic dynamics , most values of yield as expected , but there are isolated values of for which is large .( for the regression method , at these resonant points . ) these resonances are easily explained as follows : equation ( [ eq - p ] ) shows that if the fourier decomposition of the observation contains a term proportional to , then there is a resonance at where , and hence , irrespective of whether the dynamics is regular or chaotic . for the plots in fig .[ fig - cscan ] , we have calculated the asymptotic growth rate using both the regression method described in section [ sec - regress ] and the correlation method described in section [ sec - corr ] .   versus for the logistic map calculated using the regression method ( top ) and correlation method ( bottom ) .we used here data points , and equally spaced values for . left : corresponding to regular dynamics ; middle : corresponding to chaotic dynamics ; right : corresponding to chaotic but non - mixing dynamics.,title="fig : " ]  versus for the logistic map calculated using the regression method ( top ) and correlation method ( bottom ) .we used here data points , and equally spaced values for . left : corresponding to regular dynamics ; middle : corresponding to chaotic dynamics ; right : corresponding to chaotic but non - mixing dynamics.,title="fig : " ]  versus for the logistic map calculated using the regression method ( top ) and correlation method ( bottom ) .we used here data points , and equally spaced values for . left : corresponding to regular dynamics ; middle : corresponding to chaotic dynamics ; right : corresponding to chaotic but non - mixing dynamics.,title="fig : " ]   versus for the logistic map calculated using the regression method ( top ) and correlation method ( bottom ) .we used here data points , and equally spaced values for . left : corresponding to regular dynamics ; middle : corresponding to chaotic dynamics ; right : corresponding to chaotic but non - mixing dynamics.,title="fig : " ]  versus for the logistic map calculated using the regression method ( top ) and correlation method ( bottom ) .we used here data points , and equally spaced values for . left : corresponding to regular dynamics ; middle : corresponding to chaotic dynamics ; right : corresponding to chaotic but non - mixing dynamics.,title="fig : " ]  versus for the logistic map calculated using the regression method ( top ) and correlation method ( bottom ) .we used here data points , and equally spaced values for . left : corresponding to regular dynamics ; middle : corresponding to chaotic dynamics ; right : corresponding to chaotic but non - mixing dynamics.,title="fig : " ]  the occurrence of resonances for isolated values of suggests using the median of the computed values of .( we use the median rather than the mean , since the median is robust against outliers associated with resonances . )  in fig . [fig - cscan]c , is shown as a function of for where the dynamics is chaotic but not mixing on the whole interval  ] and is the noise - level in percent . fig .[ fig - noise ] shows how the undamped version of the test in this paper copes with a noise level of and the improvement that is obtained by using the damped mean - square - displacement .we obtain similar results for normally distributed noise .   versus for the logistic map increased in increments of .the darker ( red ) lines were computed using clean data .the lighter ( green ) lines were computed after addition of uniformly distributed measurement noise .both lines were computed using the undamped mean - square - displacement . left : using the undamped mean - square - displacement , right : using .bottom : using the damped mean - square - displacement with ., title="fig : " ]  versus for the logistic map increased in increments of .the darker ( red ) lines were computed using clean data .the lighter ( green ) lines were computed after addition of uniformly distributed measurement noise .both lines were computed using the undamped mean - square - displacement . left : using the undamped mean - square - displacement , right : using .bottom : using the damped mean - square - displacement with ., title="fig : " ]   versus for the logistic map increased in increments of .the darker ( red ) lines were computed using clean data .the lighter ( green ) lines were computed after addition of uniformly distributed measurement noise .both lines were computed using the undamped mean - square - displacement . left : using the undamped mean - square - displacement , right : using .bottom : using the damped mean - square - displacement with . ]  under the assumption that the noise is diffusive and not correlated with the dynamics , the mean square displacement for data contaminated with measurement noise may be written as where for a given value of , is the variance associated with the deterministic dynamics and the variance associated with the measurement noise .consider an idealized situation where the value of is roughly constant as a parameter is varied .suppose further that the dynamics is known to be regular at . then we may estimate by making a gauge - measurement at , applying the correlation method to .the unique value which yields is our estimate for . for other values of we may now apply the correlation method to .we have presented a guide for the implementation of the test for chaos . at the same time, we have introduced an improved version of the test which uses analytical expressions derived in .issues such as oversampling for continuous - time data and the presence of noise have been discussed .we hope that this guide will be helpful for scientists who would like to use the test .there are numerous methods in the literature for distinguishing between deterministic and chaotic dynamics . in our previous papers , we made a careful comparison of the test with methods for computing the maximal lyapunov exponent .another method is to use the power spectrum for which there are efficient computational techniques .it should be pointed out however that these techniques generally rely on the wiener - khintchine theorem which assumes summable decay of correlations and hence excludes periodic and quasiperiodic dynamics .hence to use power spectra as a test for chaos , it seems necessary to avoid the wiener - khintchine theorem and to work directly with the expression .equation  shows the relationship between the test and power spectra , and our test can be viewed as a way of condensing the information relevant for chaoticity or regularity contained in the power spectrum into a single binary number .we would like to thank ramon xulvi - brunet for pointing us towards the correlation method , and for explaining us the problem of oversampling in power spectra .we would like to thank michael breakspear and john dawes for providing encouraging feedback on an earlier version of the manuscript .gag is partly supported by arc grant dp0452147 and dp0667065 .the research of i m was partly supported by epsrc grant ep / d055520/1 and by a leverhulme research fellowship .i m is grateful to the university of sydney for its hospitality .m. field , i. melbourne and a. trk . decay of correlations , central limit theorems and approximation by brownian motion for compact lie group extensions ._ ergodic theory dyn .* 23 * ( 2003 ) 87110 .m. fouchard , e. lega , c. froeschl and c. froeschl . on the relationship between fast lyapunov indicator and periodic orbits for continuous flows ._ celestial mech .astronom . _* 83 * ( 2002 ) 205222 .

Abstract: in this paper we address practical aspects of the implementation of the - test for chaos in deterministic systems . in addition , we present a new formulation of the test which significantly increases its sensitivity . the test can be viewed as a method to distill a binary quantity from the power spectrum . the implementation is guided by recent results from the theoretical justification of the test as well as by exploring better statistical methods to determine the binary quantities . we give several examples to illustrate the improvement .