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The paper.y and the wrote U.S., and data, analyzed A.S., contributions: Author evidence Neurological 31). affecting (30, jointly thereby profiles inter- (29), performance practice performance and lifetime determining intelligence that in shown (28), act has practice Research of work 27). role recent the (26, and emphasized above has on or performance expert rely 50 on age crystallized that until to tasks increases (related in intelligence) knowledge performance accumulated whereas and age, experience exhibits with intelligence) fluid decline reasoning to a or (related visualization, processing are information memory, that cross- in speed, tasks of in to performance results related that primarily The shown dimensions. have studies various sectional in performance tive in profiles (25). precluded age–performance context has of this which evolution 24), long-run (23, the technology analyzing in over creativity changes performance to cycle scientific due life on time in in shifts work variation substantial Related documented and 22). has self-selection or (21, of productivity tasks measuring presence cohorts. job-related to the across related in dynamics and is performance the time problem over of short additional profile to analysis An limited age–performance an are the but prevent (20) of that y performance 44 in periods to peak 35 observation a of for ages evidence around and found potential time have changes demand over for skill account variation that in Investigations exhibit 19). (1, all groups skill which across factors, demand institutions, compara- and market of labor terms environment, work in technological limited bility, are and however, tasks, demanding c,1 owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence work.y whom this To to equally contributed D.Z. and U.S., A.S., cosbrhchrsrte hnoe time. over than dynamics rather with cohorts performance, associated birth is individual across that in ages, increase younger an at and particularly cycle configuration. life given performance the a of best over profile in hump-shaped the a engine document to chess findings The a relative per- is by players move-by-move suggested that chess per-move individual methodology professional cognitive of a of using comparison of formance y the patterns 125 on cycle past based life the over the formance long estimate the the over We particularly about tasks, run. known such is in performance little cycle workplace, life cognitively the of in importance tasks increasing demanding an for evidence Despite Significance xsigwr ncgiiepyhlg a esrdcogni- measured has psychology cognitive in work Existing y tM at ¨ cnmqePrs(NA) Institut (ENSAE), Paris economique ´ nhn 03 M 80539 unchen, ¨ . y raieCmosAttribution-NonCommercial- Commons Creative . y https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS nhn emn;and Germany; unchen, ¨ | f7 of 1

ECONOMIC SCIENCES suggests that learning and adaptability are related to biological out their entire lives and contain performance information for changes over the life cycle (32, 33). However, measures of cogni- world champions and their respective opponents. The high stakes tive performance often involve abstract tasks that are unfamiliar related to financial rewards and reputation rule out incentive to subjects and unrelated to their professional activity. More- problems. At the same time, the analysis delivers an estimate of over, they are typically only available at one point in life (e.g., the life cycle patterns of cognitive performance and its dynamics for military conscripts), which has prevented their use for study- over time and across cohorts. ing age-related variability and period–cohort decompositions The empirical strategy is based on the comparison of individ- (34). Due to the lack of direct measures of cognitive perfor- ual performance against an objective benchmark—the optimal mance that exhibit within-person variation over a sufficiently move for a given configuration suggested by a chess engine—as a large age spectrum and that are comparable across individuals measure of cognitive performance that is fully comparable over and over time, longitudinal studies for cognitive performance long time ranges. Longitudinal data for the same individuals over profiles over the life span, in particular over long horizons, are the life cycle allow disentangling age patterns from cohort and missing. period effects in a nonlinear specification. This enables an explo- This paper develops an empirical strategy to estimate the age ration of the long-run dynamics of the age–performance profile profile of performance in cognitively demanding tasks and its across periods and cohorts. In particular, the analysis applies dynamics over the past 125 y, based on the performance of flexible panel regression models to estimate the age profile of professionals in high-stakes environments related to their profes- cognitive performance and its changes across groups of birth sion. Concretely, the empirical strategy is based on the analysis of cohorts and periods (Materials and Methods). data from professional chess tournaments involving world cham- Previous work on the variation of cognitive performance in pions and their opponents. These data have several features that the context of chess either has been based on behavioral exper- make them ideal for measuring age–performance profiles and iments with a cross-section of chess players of various ages and their long-run dynamics. First, chess has been used in psychology strength, focusing on measures of decision speed and working and neuroscience as a paradigmatic cognitive task that combines memory (44), or has been based on variation between individ- processes related to perception, memory, and problem solving uals using rating information as a proxy for performance in the (35, 36). Chess has a complex neural basis of automated pro- relation between age and mental performance among amateur cesses related to identifying the configuration of pieces and their chess players (29, 30, 43, 45). In contrast, the analysis here uses relations on the board, which involve circuits of different brain within-individual variation over a long range of time and many regions (37, 38). Mounting evidence from psychology suggests cohorts and measures cognitive performance using a compara- that becoming an expert in chess and other cognitively demand- ble and objective measure based on move quality. This allows ing tasks is not just related to higher innate cognitive abilities for exploring the dynamics of the age–performance profile over but also to training and the accumulation of experience (35, a long time horizon. 39–42). The quality of a particular move thus reflects an ideal measure of performance in a demanding cognitive task of the The Dynamics of the Age–Performance Profile sort that is gaining importance in the labor market. Second, The estimation results reveal several insights about the life cycle chess data are of exceptionally high quality and allow for mea- profile of cognitive performance and its dynamics over the past suring individual performance with extreme accuracy, at the level 125 y. Fig. 1A shows the life cycle pattern of performance by plot- of individual moves during a chess game. In particular, perfor- ting the results of nonparametric estimates of the age profile mance in chess can be measured against an objective benchmark, using a local linear regression without conditioning on addi- the move that a chess engine suggests as the best-possible move tional control variables and using data for the pooled sample when facing the exact same decision problem. This allows for of world champions and their opponents. Performance reveals a constructing a measure of performance by comparing actual indi- hump-shaped pattern over the life cycle. Individual performance vidual moves with the optimal move for a given configuration. increases sharply until the early 20s and then reaches a plateau, Third, the exact same benchmark can be applied to each con- with a peak around 35 y and a sustained decline at higher ages. figuration, and the benchmark does not change over time. In The emerging life cycle performance pattern corresponds to sev- contrast to the use of ratings that change over time and with eral findings in the previous literature that have estimated the the performance of others (43), the use of move-level perfor- age profile on the basis of variation between individuals or using mance has the advantage of measuring performance in a fully work-related measures. comparable way across individuals and regardless of the tempo- To rule out that the estimate of the life cycle performance pat- ral or environmental context. This implies that performance can tern is driven by third factors, we estimated richer specifications be compared directly within individuals and across individuals, of a multivariate regression model that controls for the color as well as over long periods of time, providing a unique possibil- of chess pieces, the number of moves per player in a game (to ity to investigate the consequences of technological change and account for fatigue), and the player-specific average complexity digitization across cohorts and over time: for instance, in the con- of a game, as well as birth cohort, calendar period, and player text of the emergence of chess engines that changed education dummies. These estimates deliver similar results in terms of a and preparation facilities. Fourth, the analysis of performance hump-shaped age pattern, with performance increasing during in a task that is observed repeatedly for the same individuals young ages and decreasing during older ages. Fig. 1B illustrates allows for decomposing age patterns based on within-individual the estimated age profile for the cubic specification of age (esti- variation from variation across different cohorts and over time. mates are presented in SI Appendix, Table S1). Paralleling the Fifth, in terms of external validity, performance estimates based previous findings, the performance peak obtained with a cubic on professional chess players are likely to constitute an upper specification for age and an extensive set of control variables is bound of cognitive performance over the life cycle. The resulting at an age of around 35 y. The subsequent decline is much less measure thus provides a unique opportunity to isolate age– pronounced in the multivariate regressions. Similar results are performance patterns and analyze their dynamics over time and obtained for a specification with age bins instead of a quadratic across cohorts. specification for age (SI Appendix, Fig. S1 and Table S1). We use data from more than 24,000 chess games between These results do not account for changes in the performance 1890 and 2014 for the best players in the world, with more of chess players over the past 125 y. Unconditional estimates than 1.6 million move-by-move observations. The data are based depicted in Fig. 2A show that the average performance was on all games played by world champions in history through- substantially higher for later-born birth cohorts. The increase

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ECONOMIC SCIENCES In sum, we presented an analysis of the long-run changes in life In total, 4,294 players (20 world champions and 4,274 opponents) are cycle profiles of cognitive performance based on panel data with observed. We omit 3,422 players (5,205 observations) because fewer than repeated observations of the same individuals over their life cycle five games are observed for each of these players. The birth years of all play- using an identical task, chess, and a fully comparable perfor- ers were collected from Wikipedia. For 98% of the remaining players, who mance evaluation across individuals and over time. The evidence represent 99% of all games in the data, we were able to obtain information about the birth year. We omit 284 observations because of missing player reveals a hump-shaped performance pattern over the life cycle. birth year. Furthermore, 34 observations are omitted because the birth year Performance increased for more recent cohorts and over time. of the player is before 1836 (the birth year of the first world champion, The age profile mainly changed across cohorts rather than over Wilhelm Steinitz), and 898 observations are omitted because the player is time, with the performance increasing faster at younger ages. aged below 12 or above 65 y. The final dataset contains 42,636 observa- tions (24,379 games and 841 players). Descriptive statistics are reported in SI Materials and Methods Appendix, Table S5. Performance in Chess. Modern chess rules originated during the fifteenth For robustness checks regarding selection, we collected data on all games century in Italy and Spain and have essentially not changed since the early of the opponents of the world champions throughout their lives from the nineteenth century. Chess is a two-person zero-sum game with perfect infor- chess database Chessbase. In contrast to the baseline data, which contain all mation and alternating moves. For this class of games, the optimal strategy games played by world champions, the alternative data sample one random is strictly determined and may be found by backward induction (50–52). For white game and one random black game for each year in which at least one each configuration, chess engines compute an evaluation that represents a game is observed in the database for a given opponent player in the baseline proxy of the winning odds and is measured in so-called pawn units, where dataset. This sample contains the same set of players as the baseline sam- one unit approximates the advantage of being up one pawn. Based on a ple but is not selected based on whether a world champion participated in a game tree for all possible moves by white/black of a given prespecified game. Performance evaluation for this dataset of 57,321 game–player obser- length of n moves ahead (the so-called search depth), engines determine vations over the period from 1890 to 2013 with 2.5 million configurations was the best next move, applying a best-response logic (SI Appendix, Fig. S23 conducted in a comparable way as in the baseline sample. shows an illustration). As a measure of performance, we compute the dif- ference in the evaluation before a move (conditional on following the Measure of Performance. Evaluations of configurations and quality of moves computed continuation path of best responses) and right after (when the were carried out with the use of STOCKFISH 8, an open-source program engine has recalculated the evaluation of the configuration). This procedure that computes, for a given configuration of the pieces on the chessboard, is equivalent to comparing the player’s positional evaluation right after the the best possible move (details are in ref. 53). With an estimated Elo- actual move with the player’s evaluation had the player conducted the move rating exceeding 3,200 points, this engine provides a relevant benchmark suggested by the engine. even for the best players in history (incumbent World Champion Magnus The central advantage of this setup for addressing the research question Carlsen had an Elo-number of 2,872 in January 2020; https://ratings. on performance over the life cycle and its long-run dynamics is that each fide.com/toparc.phtml?cod=577; last accessed 17 March 2020). move played in the dataset can be evaluated using the exact same objective In game g, the evaluation of player i’s position according to the chess benchmark, regardless of period, age, or birth cohort of the player. Alterna- engine is Eigm0 pawn units before move m and Eigm pawn units after move tive measures such as conventional Elo-ratings (43) are not comparable over m, so that the corresponding change of the evaluation as a result of move m time. is ∆igm = Eigm − Eigm0 pawn units. When computing Eigm0 , the chess engine assumes that the player would play the move that the chess engine evalu- Data. The data are a collection of all games played at regular time ates as optimal. This implies that ∆igm = 0 if a player plays the optimal move controls (usually 40 moves within 2 h) by all chess world champions according to the chess engine and ∆igm < 0 if the player plays a move that since the first generally accepted world champion Wilhelm Steinitz (lived the engine evaluates as suboptimal. Thus, ∆igm is an increasing measure of 1836 to 1900) to Magnus Carlsen (born in 1990, world champion since performance of player i for move m in game g. This performance measure 2013). The data were assembled originally in ref. 53 and are based on is comparable across different configurations, moves, games, and players the commercially available chess database Chessbase and other online because it is benchmarked against the objective computer-generated move. sources commonly used in the chess community (the data are available Players can achieve the performance of the computer-generated bench- at http://www.alliot.fr/CHESS/ficga.html.en). The main dataset comprises mark independent of endogenous factors, such as the initial evaluation of 25,072 games with more than 1.6 million configurations, which were played the configuration, the strength of the opponent, or the complexity of the by (or against) the world champions of chess since 1859.† The data con- configuration on the board. tain nearly the entire lifetime history of games played in chess competitions As a baseline measure of performance, we use the share of optimal moves by world champions (including games before and after their acting as world of a player in a given game. This measure reflects an overall composite of champion). The dataset contains detailed information about the date of the performance and is computed as game, color of chess pieces, scored points, and chessboard configurations before and after each move (of the world champions and their opponents). #(i,g) ¯ 1 X 
To rule out memorized moves and economize on computing costs, only Pig = 1 − I ∆igm < 0 · 100%, #(i, g) moves between moves 10 and 100 (so-called “out-of-book moves”) are m=1 considered (the first moves of a chess game, “book moves,” are studied intensively during the preparation of a game and usually correspond to with #(i, g) being the total number of moves of player i in game g and routine openings that have been memorized by players). I{·} being the indicator function. The resulting measure is distributed sym- After generating the move-by-move measure for performance, the data metrically between 0 and 100 and correlates positively with the probability ‡ were aggregated on the player–game level. This implies two observations of winning as reflected by scoring points (SI Appendix, Fig. S24). A greater per game, one for each player. The aggregated data contain 50,143 game– share of optimal moves is associated with a higher winning probability, with player observations (for one game by Tigran Petrosian, we do not observe more than 40% of optimal moves effectively implying a winning probability the moves of the opponent; we drop this game because it lasted for fewer of more than 50%. As an alternative measure, we use the average (logarith- than 20 moves); 329 observations are omitted because the respective game mic) distance between actual moves and the computer-generated optimal lasted for fewer than 20 moves. Furthermore, we restrict attention to games benchmark in terms of pawn units (SI Appendix, Fig. S25). played since 1890, which implies that 757 observations of games that took In addition, the data contain information about the search depth in terms place before 1890 are omitted. of nodes that the chess engine was able to evaluate within a prespecified time limit of 3 min. Since this search depth is directly related to the branch- ing factor of the game tree that arises from a given chess configuration (as illustrated in SI Appendix, Fig. S23), lower search depth provides a use- †They are Wilhelm Steinitz, Emanuel Lasker, Jose´ Raul´ Capablanca, Alexander Alekhine, ful measure of the complexity of each configuration. We aggregated this Max Euwe, Mikhail Botvinnik, Vasily Smyslov, Mikhail Tal, Tigran Petrosian, Boris measure on the game level to obtain a measure of the average complexity Spassky, Robert James Fischer, Anatoly Karpov, Gary Kasparov, Alexander Khalifman, of each game. Viswanathan Anand, Ruslan Ponomariov, Rustam Kasimdzhanov, Veselin Topalov, Vladimir Kramnik, and Magnus Carlsen. Empirical Strategy. The empirical analysis is based on the partial linear ‡The move-by-move evaluation follows that by Alliot (53). model,

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Charness, N. 41. atcpnsa h KLSmnr nebr,ada h iia Economy and Digital Ours, the comments. van at helpful for Jan and University Aviv Alex Engelberg, Stillman, Tel Fitzenberger, SKIL-Seminar, at Steve Ludwig-Maximilians- Workshop Bernd the Lalive, Ericsson, at at Rafael Anders participants Studies Ichino, K. Andrea Biroli, Economic Krumer, Pietro for thank authors Center the Universit 190). thanks TRR (CRC 280092119 A.S. and 395413683 Projects Forschungsgemeinschaft ACKNOWLEDGMENTS. calendar (https://doi.org/10.7910/DVN/DZC0MT). 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