Artificial intelligence (AI) has achieved human-like performance in a growing number of tasks, such as visual recognition and natural language processing.111The formal definition of AI seems contentious, partly because scholars have not agreed on the definition of intelligence in the first place. This paper follows a broad definition of AI as computer systems able to perform tasks that traditionally required human intelligence.
The classical games of chess, shogi (Japanese chess), and Go were once thought to be too complicated and intractable for AI, but computer scientists have overcome these challenges. In chess, IBM’s computer system named Deep Blue defeated Grandmaster Garry Kasparov in 1997. In shogi, a machine-learning-based program called Bonanza challenged (and was defeated by) Ryūō champion Akira Watanabe in 2007, but one of its successors (Ponanza) played against Meijin champion Amahiko Satoh and won in 2017. In Go, Google DeepMind developed AlphaGo, a deep-learning-based program, which beat the 2-dan European champion Fan Hui in 2015, a 9-dan (highest rank) professional Lee Sedol in 2016, and the world’s best player Ke Jie in 2017.
Despite such remarkable achievements, one of the lingering criticisms of AI is its lack of transparency. The internal mechanism seems like a black box to most people, including the human experts of the relevant tasks,222For example, Yoshiharu Habu, the strongest shogi player in recent history, states he does not understand certain board-evaluation functions of computer shogi programs (Habu and NHK ). which raises concerns about accountability and responsibility. The desire to understand and explain the functioning of AI is not limited to the scientific community. For example, the US Department of Defense airs its concern that “the effectiveness of these systems is limited by the machine’s current inability to explain their decisions and actions to human users,” which led it to host the Explainable AI (XAI) program aimed at developing “understandable” and “trustworthy” machine learning.333See https://www.darpa.mil/program/explainable-artificial-intelligence (accessed on October 17, 2017).
This paper examines three prominent game AIs in recent history: Deep Blue, Bonanza, and AlphaGo. I have chosen to study this category of AIs because board games represent an archetypical task that has required human intelligence, including cognitive skills, decision-making, and problem-solving. They are also well-defined problems for which economic interpretations are more natural than for, say, visual recognition and natural language processing. The main finding from this paper’s case studies is that these AIs’ key components are mathematically equivalent to well-known econometric methods to estimate dynamic structural models.
Chess experts and IBM’s engineers manually adjusted thousands of parameters in Deep Blue’s “evaluation function,” which quantifies the probability of eventual winning as a function of the current positions of pieces (i.e., state of the game) and therefore could be interpreted as an approximate value function. Deep Blue is a calibrated value function with a linear functional form.
By contrast, the developer of Bonanza constructed a dataset of professional shogi games, and used a discrete-choice regression and a backward-induction algorithm to determine the parameters of its value function. Hence, his method of “supervised learning” is equivalent to Rust’s (1987) nested fixed-point (NFXP) algorithm, which combined a discrete-choice model with dynamic programming (DP) in the maximum likelihood estimation (MLE) framework. Bonanza is an empirical model of human shogi players that is estimated by this direct (or “full-solution”) method.
Google DeepMind’s AlphaGo (its original version) embodies an alternative approach to estimating dynamic structural models: two-step estimation.444This paper focuses on the original version of AlphaGo, published in 2016, and distinguishes it from its later version, “AlphaGo Zero,” published in 2017. The latter version contains few econometric elements, and is not an immediate subject of my case study, although I discuss some of its interesting features in section 5. Its first component, the “supervised-learning (SL) policy network,” predicts the moves of human experts as a function of the board state. It is an empirical policy function with a class of nonparametric basis functions (DNN: deep neural network) that is estimated by MLE, using data from online Go games. Thus, the SL policy network is a DNN implementation of Hotz and Miller’s (1993) first-stage conditional choice probability (CCP) estimation.
AlphaGo’s value function, called “reinforcement-learning (RL) value network,” is constructed by simulating many games based on the self-play of the SL policy network and estimating another DNN model that maps state to the probability of winning. This procedure is equivalent to the second-stage conditional choice simulation (CCS) estimation, proposed by Hotz, Miller, Sanders, and Smith (1994) for single-agent DP, and by Bajari, Benkard, and Levin (2007) for dynamic games.
Thus, these leading game AIs and the core algorithms for their development turn out to be successful applications of the empirical methods to implement dynamic structural models. After introducing basic notations in section 2, I describe the main components of Deep Blue, Bonanza, and AlphaGo in sections 3, 4, and 5, respectively, and explain their structural interpretations. Section 6 clarifies some of the implicit assumptions underlying these AIs, such as (the absence of) unobserved heterogeneity, strategic interactions, and various constraints human players are facing in real games. Section 7 concludes by suggesting that relaxing some of these assumptions and explicitly incorporating more realistic features of the data-generating process could help make AIs both more human-like (if needed) and more amenable to structural interpretations.
Literature This paper clarifies the equivalence between some of the algorithms for developing game AI and the aforementioned econometric methods for estimating dynamic models. As such, the most closely related papers are Rust (1987), Hotz and Miller (1993), and Hotz, Miller, Sanders, and Smith (1994). The game AIs I analyze in this paper are probably the most successful (or at least the most popular) empirical applications of these methods. For a historical review of numerical methods for dynamic programming, see Rust (2017).
At a higher level, the purpose of this paper is to clarify the connections between machine learning and econometrics in certain areas. Hence, the paper shares the spirit of, for example, Belloni, Chernozhukov, and Hansen (2014), Varian (2014), Athey (2017), and Mullainathan and Spiess (2017), among many others in the rapidly growing literature on data analysis at the intersection of computer science and economics.
Chess, shogi, and Go belong to the same class of games, with two players (), discrete time (), alternating moves (players 1 and 2 choose their actions,
, in odd and even periods, respectively), perfect information, and deterministic state transition,
where both the transition, , and the initial state, , are completely determined by the rule of each game.555This setup abstracts from the time constraints in official games because the developers of game AIs typically do not incorporate them at the data-analysis stage. Hence, represents turn-to-move, not clock time. Section 6 investigates this issue.
Action space is finite and is defined by the rule as “legal moves,”
State space is finite as well, and consists of four mutually exclusive subsets:
where I denote “win” and “loss” from the perspective of player 1 (e.g., player 1 wins and player 2 loses if ). Neither player wins if . The game continues as long as .
The two players’ payoffs sum to zero:
with defined in a similar manner (but with win/loss payoffs flipped). This setup means chess, shogi, and Go are well-defined finite games. In principle, such games can be solved exactly and completely by backward induction from the terminal states.
In practice, even today’s supercomputers and a cloud of servers cannot solve them within our lifetime, because the size of the state space, , is large. The approximate of chess, shogi, and Go are , , and , respectively, which are comparable to the number of atoms in the observable universe () and certainly larger than the total information-storage capacity of humanity (in the order of bytes).666In 2016, the world’s hard disk drive (HDD) industry produced a total of 693 exabytes (EB), or bytes.
3 Chess: Deep Blue
IBM’s Deep Blue is a computer system with custom-built hardware and software components. I focus on the latter, programming-related part. Deep Blue’s program consists of three key elements: an evaluation function, a search algorithm, and databases.
The “evaluation function” of Deep Blue is a linear combination of certain features of the current board state . It quantifies the probability of eventual winning () or its monotonic transformation, :
is a vector ofparameters and is a vector of observable characteristics of . The published version featured parameters (Campbell, Hoane, and Hsu ).
A typical evaluation function for computer chess considers the “material value” associated with each type of piece, such as 1 point for a pawn, 3 points for a knight, 3 points for a bishop, 5 points for a rook, 9 points for a queen, and an arbitrarily many points for a king (e.g., 200 or 1 billion), because the game ends when a king is captured. Other factors include the relative positions of these pieces, such as pawn structure, protection of kings, and experts’ opinion that a pair of bishops are usually worth more than the sum of their individual material values. Finally, the importance of these factors may change depending on the phase of the game: opening, middle, or endgame.
Reasonable parameterization and the choice of board characteristics (variables) require expert knowledge. Multiple Grandmasters (the highest rank of chess players) advised the Deep Blue development team. However, they did not use statistical analysis or data from professional players’ games. In other words, they did not estimate . Each of the 8,150 parameters, , was manually adjusted until the program’s performance reached a satisfactory level.
The second component of Deep Blue is “search,” or a solution algorithm to choose the optimal action at each turn to move. In its “full-width search” procedure, the program evaluates every possible position for a fixed number of future moves along the game tree, using the “minimax algorithm” and some “pruning” methods. This “search” is a version of numerical backward induction.
Deep Blue uses two databases: one for the endgame and the other for the opening phase.
The endgame database embodies the cumulative efforts by the computer chess community to solve the game in an exact manner. Ken Thompson and Lewis Stiller developed Deep Blue’s endgame database with all five-piece positions (i.e., the states with only five pieces, and all possible future states that can be reached from them), as well as selected six-piece positions.777A database with all five-piece endings requires 7.05 gigabytes of hard disk space; storing all six-piece endings requires 1.2 terabytes.
The second database is an “opening book,” which is a collection of common openings (i.e., move patterns at the beginning of the game) experts consider good plays. It also contains good ways to counter the opponent’s poor openings, again based on the judgment by experts. Grandmasters Joel Benjamin, Nick De Firmian, John Fedorowicz, and Miguel Illescas created one with about 4,000 positions, by hand.
The team also used some data analysis to prepare an “extended book” to guard against non-standard opening positions. Campbell, Hoane, and Hsu’s (2002) description suggests they constructed a parametric move-selection function based on the data analysis of 700,000 human games. They even incorporated annotators’ commentary on the moves. Nevertheless, the use of data analysis seems limited to this back-up database.
Deep Blue lost a match to the top-ranked Grandmaster Garry Kasparov in 1996, but defeated him in 1997. Since then, the use of computer programs has become widespread in terms of both training and games (e.g., those played by hybrid teams of humans and computers).888See Kasparov (2007), for example.
3.2 Deep Blue Is a Calibrated Value Function
The fact that IBM manually adjusted the parameter by trial and error means Deep Blue is a fruit of painstaking efforts to “calibrate” a value function with thousands of parameters: Deep Blue is a calibrated value function.
A truly optimal value function would obviate the need for any forward-looking and backward-induction procedures to solve the game (i.e., the “search algorithm”), because the true value function should embody such solution. However, the parametric value function is merely an attempt to approximate the optimal one. Approximation errors (or misspecification) seem to leave room for performance improvement by the additional use of backward induction, although such benefits are not theoretically obvious.
The “full-width search” procedure is a brute-force numerical search for the optimal choice by backward induction, but solving the entire game is infeasible. Hence, this backward induction is performed on a truncated version of the game tree (truncated at some length from the current turn ). The “terminal” values at turn are given by the parametric value function (5). The opponent is assumed to share the same terminal-value function at and to choose its move to minimize the focal player’s .
Thus, Deep Blue is a parametric (linear) function to approximate the winning probability at the end of a truncated game tree, , in which the opponent shares exactly the same value function and the time horizon. In other words, Deep Blue is a calibrated, approximate terminal-value function in a game the program plays against its doppelgänger.
4 Shogi: Bonanza
In 2005, Kunihito Hoki, a chemist who specialized in computer simulations at the University of Toronto, spent his spare time developing a shogi-playing program named Bonanza, which won the world championship in computer shogi in 2006. Hoki’s Bonanza revolutionized the field of computer shogi by introducing machine learning to “train” (i.e., estimate) a more flexible evaluation function than either those for chess or those designed for the existing shogi programs.999Hoki acknowledges machine-learning methods had previously been used in computer programs to play Backgammon and Reversi (“Othello”), but says he could not find any successful applications to chess or shogi in the literature (Hoki and Watanabe ).
More Complicated Evaluation Function
The developers at IBM could manually adjust 8,150 parameters in and beat the human chess champions. The same approach did not work for shogi. Shogi programs before Bonanza could only compete at amateurs’ level at best. This performance gap between chess and shogi AIs is rooted in the more complicated state space of shogi, with .
Several factors contribute to the complexity of shogi: a larger board size (), more pieces (), and more types of pieces (). In addition, most of the pieces have limited mobility,101010Four of the eight types of pieces in shogi can move only one unit distance at a time, whereas only two of the six types of pieces in chess (pawn and king) have such low mobility. The exact positions of pieces becomes more important in characterizing the state space when mobility is low, whereas high mobility makes pure “material values” relatively more important, because pieces can be moved to wherever they are needed within a few turns. and, other than kings, never die. Non-king pieces are simply “captured,” not killed, and can then be “dropped” (re-deployed on the capturer’s side) almost anywhere on the board at any of the capturer’s subsequent turns. This last feature is particularly troublesome for an attempt to solve the game exactly, because the effective does not decrease over time.
Hoki designed a flexible evaluation function by factorizing the positions of pieces into (i) the positions of any three pieces including the kings and (ii) the positions of any three pieces including only one king. This granular characterization turned out to capture important features of the board through the relative positions of three-piece combinations. Bonanza’s evaluation function, , also incorporated other, more conventional characteristics, such as individual pieces’ material values (see Hoki and Watanabe , pp. 119–120, for details). As a result, has a linear functional form similar to (5) but contains variables and the same number of parameters (Hoki ).
Machine Learning (Logit-like Regression)
That the Deep Blue team managed to adjust thousands of parameters for the chess program by human hands is almost incredible. But the task becomes impossible with 50 million parameters. Hoki gave up on manually tuning Bonanza’s and chose to use statistical methods to automatically adjust based on the data from the professional shogi players’ 50,000 games on official record: supervised learning.
Each game takes 100 moves on average. Hence, the data contain approximately 5 million pairs of .111111In earlier versions of Bonanza, Hoki also used additional data from 50,000 unofficial, online game records as well, to cover some rare states such as nyuu-gyoku positions (in which a king enters the opponent’s territory and becomes difficult to capture, because the majority of shogi pieces can only move forward, not backward). However, he found the use of data from amateur players’ online games weakened Bonanza’s play, and subsequently abandoned this approach (Hoki ). The reader might notice the sample size is smaller than (50 million). Hoki reduced the effective number of parameters by additional restrictions to “stabilize the numerical optimization process.”
Like Deep Blue, Bonanza chooses its move at each of its turns by searching for the action that maximizes in some future turn ,
assuming the opponent shares the same terminal-value function and tries to
minimize it. Note the “optimal” choice is inherently related to through the objective
function . This relationship can be
exploited to infer from the data on .
Hoki used some variant of the discrete-choice regression method to determine
the values of .121212 Tomoyuki Kaneko states he also used some machine-learning methods as early
as 2003 for his program named GPS Shogi (Kaneko ). Likewise, Yoshimasa
Tsuruoka writes he started using logit regressions in 2004 for developing
his own program, Gekisashi (Tsuruoka ). But shogi programmers seem to
agree that Hoki’s Bonanza was the first to introduce data-analysis methods
for constructing an evaluation function in a wholesale manner.
Tomoyuki Kaneko states he also used some machine-learning methods as early as 2003 for his program named GPS Shogi (Kaneko ). Likewise, Yoshimasa Tsuruoka writes he started using logit regressions in 2004 for developing his own program, Gekisashi (Tsuruoka ). But shogi programmers seem to agree that Hoki’s Bonanza was the first to introduce data-analysis methods for constructing an evaluation function in a wholesale manner.
Bonanza won the world championship in computer shogi in 2006 and 2013. In 2007, the Ryūō (“dragon king,” one of the two most prestigious titles) champion Akira Watanabe agreed to play against Bonanza and won. After the game, however, he said he regretted agreeing to play against it, because he felt he could have lost with non-negligible probabilities. Hoki made the source code publicly available. The use of data and machine learning for computer shogi was dubbed the “Bonanza method” and copied by most of the subsequent generations of shogi programs.
Issei Yamamoto, a programmer, named his software Ponanza out of respect for the predecessor, claiming his was a lesser copy of Bonanza. From 2014, Ponanza started playing against itself in an attempt to find “stronger” parameter configurations than those obtained (estimated) from the professional players’ data: reinforcement learning (Yamamoto ). Eventually, Ponanza became the first shogi AI to beat the Meijin (“master,” the other most prestigious title) champion in 2017, when Amahiko Satoh lost two straight games.
4.2 Structural Interpretation: Bonanza Is Harold Zurcher
Bonanza is similar to Deep Blue. Its main component is an approximate terminal-value function, and the “optimal” action is determined by backward induction on a truncated game tree of self play (equation 6). The only difference is the larger number of parameters (50 million), which reflects the complexity of shogi and precludes any hopes for calibration. Hence, Hoki approached the search for as a data-analysis problem.
Accordingly, Bonanza is an empirical model of professional shogi players, in the same sense that Rust (1987) is an empirical model of Harold Zurcher, a Madison, Wisconsin, city-bus superintendent. Rust studied his record of engine-replacement decisions and estimated his utility function, based on the principle of revealed preference. This comparison is not just a metaphor. The machine-learning algorithm to develop Bonanza is almost exactly the same as the structural econometric method to estimate an empirical model of Harold Zurcher.
Rust’s (1987) “full-solution” estimation method consists of two numerical optimization procedures that are nested. First, the overall problem is to find that makes the model’s prediction (as a function of and ) fit the observed action-state pairs in the data . Second, the nested sub-routine takes particular as an input and solves the model to find the corresponding policy function (optimal strategy),
The first part is implemented by the maximum likelihood method (i.e., the fit is evaluated by the proximity between the observed and predicted choice probabilities). The second part is implemented by value-function iteration, that is, by numerically solving a contraction-mapping problem to find a fixed point, which is guaranteed to exist and is unique for a well-behaved single-agent dynamic programming (DP) problem. This algorithm is called nested fixed-point (NFXP) because of this design.
Hoki developed Bonanza in the same manner. The overall problem is to find that makes Bonanza predict the human experts’ actions in the data (7). The nested sub-routine takes as an input and numerically searches for the optimal action by means of backward induction. The first part is implemented by logit-style regressions (i.e., the maximum-likelihood estimation of the discrete-choice model in which the error term is assumed i.i.d. type-1 extreme value). This specification is the same as Rust’s. The second part proceeds on a truncated game tree, whose “leaves” (i.e., terminal values at ) are given by the approximate value function , and the opponent is assumed to play the same strategy as itself:
Strictly speaking, Bonanza differs from (the empirical model of) Harold Zurcher in two aspects. Bonanza plays a two-player game with a finite horizon, whereas Harold Zurcher solves a single-agent DP with an infinite horizon. Nevertheless, these differences are not as fundamental as one might imagine at first glance, because each of them can be solved for a unique “optimal” strategy in the current context. An alternating-move game with a finite horizon has a unique equilibrium when i.i.d. utility shock is introduced and breaks the tie between multiple discrete alternatives. Igami (2017, 2018) demonstrates how Rust’s NFXP naturally extends to such cases with a deterministic order of moves; Igami and Uetake (2017) do the same with a stochastic order of moves. Thus, Bonanza is to Akira Watanabe what Rust (1987) is to Harold Zurcher.
5 Go: AlphaGo
The developers of AIs for chess and shogi had successfully parameterized state spaces and constructed evaluation functions. Meanwhile, the developers of computer Go struggled to find any reasonable parametric representation of the board.
Go is more complicated than chess and shogi, with . Go has only one type of piece, a stone, and the goal is to occupy larger territories than the opponent when the board is full of black and white stones (for players 1 and 2, respectively). However, the board size is much larger, and so is the action space. Practically all open spaces constitute legal moves. The local positions of stones seamlessly interact with the global ones. Even the professional players cannot articulate what distinguishes good positions from bad ones, frequently using phrases that are ambiguous and difficult to codify. The construction of a useful evaluation function was deemed impossible.
Instead, most of the advance since 2006 had been focused on improving game-tree search algorithms (Yoshizoe and Yamashita , Otsuki ). Even though the board states in the middle of the game are difficult to codify, the terminal states are unambiguous, with either win or loss. Moreover, a “move” in Go does not involve moving pieces that are already present on the board; it comprises simply dropping a stone on an open space from outside the board. These features of Go make randomized “play-out” easy. That is, the programmer can run Monte Carlo simulations in which black and white stones are alternatingly dropped in random places until the board is filled. Repeat this forward simulation many times, and one can calculate the probability of winning from any arbitrary state of the board . This basic idea is behind a method called Monte Carlo tree search (MCTS).
Given the large state space, calculating the probability of winning (or its monotonic transformation ) for all ’s remains impractical. However, a computer program can use this approach in real time to choose the next move, because it needs to compare only alternative actions and their immediate consequences (i.e., states) at its turn to move. Forward simulations involve many calculations, but each operation is simple and parallelizable. That is, computing one history of future play does not rely on computing another history. Likewise, simulations that start from a particular state do not have to wait for the completion of other simulations that start from , where . Such computational tasks can be performed simultaneously on multiple computers, processors, cores, or GPUs (graphic processing units). If the developer can use many computers during the game, the MCTS-based program can perform sufficiently many numerical operations to find good moves within a short time.
Thus, MCTS was the state of the art in computer Go programming when Demis Hassabis and his team at Google DeepMind proposed a deep-learning-based AI, AlphaGo. The four components of AlphaGo are (i) a policy network, (ii) RL, (iii) a value network, and (iv) MCTS. Among these four components, (i) and (iii) were the most novel features relative to the existing game AIs.
Component 1: Supervised Learning (SL) of the Policy Network
The first component of AlphaGo is a “policy network,” which is a deep neural network (DNN) model to predict professional players’ move as a function of current state . It is a policy function as in (7), with a particular functional form and 4.6 million “weights” (i.e., parameter vector ).131313I use to denote the parameter vector of policy function, and distinguish it from the parameter vector of the value function .
Like Hoki did for Bonanza, the AlphaGo team determined by using the data from an online Go website named Kiseido Go Server (KGS). Specifically, they used the KGS record on 160,000 games played by high-level (6-9 dan) professionals. A game lasts 200 moves on average, and eight symmetric transformations (i.e., rotations and flipping) of the board generate nominally different states. Hence, the effective size of the dataset is
Note the sample size is still small (negligible) relative to .
Its basic architecture is a standard Convolutional Neural Network (CNN), which is known to perform well in image-recognition tasks, among others. The Appendix describes the details of AlphaGo’s DNN functional-form specification. Its final output is the prediction of choice probabilities across all legal moves based on the following logit formula:
where indexes actions and is the deterministic part of the latent value of choosing action in state given parameter .
This specification is “deep” in the sense that the model contains multiple layers, through which
is calculated. It is named “neural network” because the layers contain many units of simple numerical operations (e.g., convolution and zero-truncation), each of which transmits inputs and outputs in a network-like architecture, with the analogy of computational nodes as biological neurons that transmit electric signals.
The approximate number of parameters in AlphaGo’s policy network is
where each of the “kernels” is a (or ) matrix that is designed to indicate the presence or absence of a particular local pattern, in each (or ) part of the board. Note the number of parameters of AlphaGo’s policy function is smaller than the 50 million parameters in Bonanza, despite the fact that Go has a larger state space than shogi.
The supervised learning (i.e., estimation) of uses a standard numerical optimization algorithm to maximize the likelihood function that aggregates the optimal choice probabilities implied by the model, the data, and the parameter values. That is, AlphaGo’s policy function is estimated by the classical maximum likelihood method. The team did not add any “regularization” term in the objective function, which is a common practice in machine learning to improve the out-of-sample prediction accuracy at the expense of biased estimates. Nevertheless, the estimated policy function, , could predict 55.7% of the human players’ moves outside the sample, and its top-five move predictions contained the actual human choices almost 90% of the time.141414By contrast, a simple parametric (logit without a DNN inside) version of the empirical policy function (for the MCTS purposes) achieved only 27% accuracy, which is still remarkable but less impressive than the DNN version’s performance.
Component 2: Reinforcement Learning (RL) of the Policy Network
The ultimate goal of the AlphaGo team was the creation of a strong AI, not the prediction of human play per se
(or the unbiased estimation of). The second ingredient of AlphaGo is the process of reinforcement learning to make a stronger policy function than the estimated one from the previous step, .
Reinforcement learning is a generic term to describe a numerical search for “better” actions based on some performance criteria, or “reward,” such as the average score of the game. The specific task in the current case is to find some such that the winning probability is higher under strategy than :
where is the probability that player wins with strategy against the opponent who uses , in terms of the average performance across many simulated plays of the game.
Because the outcome of the game depends on both and , condition (10) does not guarantee the superiority of in general (i.e., when playing against any strategies other than ). The only way to completely address this issue is to solve the game exactly for the optimal strategy , but such a solution is computationally impossible. Accordingly, the development team tries to find “satisficing” , by making each candidate policy play against many different policies that are randomly sampled from the previous rounds of iteration (i.e., various perturbed versions of in the numerical search process), and by simulating plays from a wide variety of that are also randomly sampled from those in the data (as well as those from perturbed versions of such historical games).
Component 3: SL/RL of Value Network
The third ingredient of AlphaGo is the evaluation function, , to assess the probability of winning from any . Constructing such an object had been deemed impossible in the computer Go community, but the team managed to construct the value function from the policy function through simulations. Specifically, they proceeded as follows:
Simulate many game plays between the RL policy function, , and itself.
Pick many (30 million) different states from separate games in the simulation and record their winners, which generates a synthetic dataset of 30 million pairs.
Use this dataset to find the value function that predicts from any .
In other words, strategy implies certain outcomes in the game against itself, and these outcomes become explicit through simulations:
Once the outcomes become explicit, the only remaining task is to fit some flexible functional form to predict as a function of , which is a plain-vanilla regression (supervised learning) task.
The developers prepared another DNN (CNN) with a design that is similar to the policy network: 49 channels, 15 layers, and 192 kernels. The only differences are an additional state variable that reflects the identity of the player (to attribute the win/loss outcome to the correct player) and an additional computational step at the end of the hierarchical architecture that takes all arrays of intermediate results as inputs and returns a scalar () as an output.
Let us denote the estimated value network by
where the expression clarifies the dependence of on the use of specific strategy by both the focal player and the opponent in the simulation step to calculate . These notations are cumbersome but help us keep track of the exact nature of the estimated value network when we proceed to their structural interpretation.
Component 4: Combining Policy and Value with MCTS
The fourth component of AlphaGo is MCTS, the stochastic solution method for a large game tree. When AlphaGo plays actual games, it combines the RL policy and the RL value (equation 12) within an MCTS algorithm.
Each of these components can be used individually, or in any combinations (Silver et al. , Figure 4). The policy function directly proposes the optimal move from any given state. The value function indirectly suggests the optimal move by comparing the winning probabilities across the next states that result from candidate moves. An MCTS can perform a similar state-evaluation task by simulating the outcomes of the candidate moves. They represent more or less the same concept: approximate solutions to an intractable problem that is guaranteed to have a unique exact solution. Nevertheless, positive “ensemble effects” from combining multiple methods are frequently reported, presumably because different types of numerical approximation errors may cancel out each other.151515This ensemble part involves many implementation details of purely computational tasks, and hence is beyond the scope of this paper.
5.2 Structural Interpretation: AlphaGo Is Two-step Estimation
Component 1: SL Policy Network Is First-Stage CCP Estimates
Deep Blue and Bonanza embody parametric value functions, whereas AlphaGo’s first key component (SL policy network) is a nonparametric policy function, which is equivalent to the estimation of CCPs in Hotz and Miller’s (1993) two-step method.
In case the reader is unfamiliar with the literature on dynamic structural models, I provide a brief summary. The NFXP method (see section 4.2) requires the solution of a DP problem, which becomes computationally expensive as the size of the state space increases. Moreover, this solution step has to be repeated for each candidate vector of parameter values .
Hotz and Miller (1993) proposed an estimation approach to circumvent this problem. To the extent that the actual choices in the data reflect the optimal choice probabilities that are conditional on the observed state in the data, we can estimate the policy function directly from the data. This procedure is the estimation of CCPs in their first stage. The benefit of this approach is that the procedure does not require solving a fully dynamic model. The cost of this approach is that the requirement for data becomes more demanding.
One should avoid imposing parametric assumptions on the first-stage policy function, because a typical goal of empirical analysis is to find the parameters of the value function (and its underlying structural components, e.g., preference and technology), , that are implied by observed actions in data and not by their parametric surrogates. A priori restrictions on the policy function could potentially contradict the solution of the underlying DP. Thus, being nonparametric and preserving flexible functional forms are crucial for an adequate implementation of the Hotz-Miller method. In this sense, the CCP method is demanding of data. It trades the computationalcurse of dimensionality for the data curse of dimensionality.
Given this econometric context, the use of DNN seems a sensible choice of functional form. In one of the foundational works for deep learning, econometrician Halbert White and his coauthors proved such a multi-layer model with many nodes can approximate any arbitrary functions (Hornik, Stinchcombe, and White ), as long as the network is sufficiently large and deep (i.e., has a sufficient degree of flexibility) to capture complicated data patterns.161616See also Chen and White (1999). For an overview on deep learning, see Goodfellow, Bengio, and Courville (2016), for example. The sole purpose and requirement of Hotz and Miller’s first stage is to capture the actual choice patterns in the data as flexibly as possible. Silver et al. (2016) report SL policy network’s out-of-sample move-prediction accuracy is 55% (and close to 90% with top-five predictions in Maddison et al. ), whereas that of a simple parametric (logit) version is 27%. This level of fit is a remarkable achievement, because the sample size is small (practically zero) relative to the size of the state space.
Component 2: RL Policy Network Is Like a “Counterfactual” with Long-Lived Players
The making of the RL policy network does not involve raw data. Rather, it is a pure numerical search for a better approximation of the truly optimal solution of the game (which is known to exist and be unique). Although this paper focuses on the original version of AlphaGo, discussing its pure-RL version (AlphaGo Zero) might be useful for a comparison:
In the case of (original) AlphaGo, RL starts from the top human players’ strategy as an initial value, and iteratively searches for a “stronger” strategy .
In the case of AlaphGo Zero, RL starts from tabula rasa (i.e., nothing but purely random play).171717See Silver et al. (2017).
Regardless of the choice of the initial value, RL in this context is a policy-function iteration (or best-response iteration) approach that has been used in many economic applications, such as Pakes and McGuire’s (1994, 2001) implementation of dynamic oligopoly games.
In the case of (original) AlphaGo, the resulting strategy could be interpreted as an outcome of some “counterfactual” experiment in which the top human players (as embodied and immortalized in ) lived long careers and accumulated additional experience. By the same token, the RL for AlaphGo Zero has a flavor of simulating the learning trajectory of a first-time player without any teacher or textbook, although the exact form of human learning from the actual games and training would be quite different.
Component 3: SL/RL Value Network Is Second-Stage CCS Estimates
According to Silver et al. (2016), AlphaGo’s SL/RL value network is the first successful evaluation function for Go, which is a remarkable achievement. The procedure to obtain this value function is a straightforward application of Hotz, Miller, Sanders, and Smith’s (1994, henceforth, HMSS) CCS estimator, combined with another DNN to approximate the complicated relationship between and in the high-dimensional state space.
The literature context is the following. Hotz and Miller (1993) proved the existence of a one-to-one mapping between the policy function and the value function, so that the former can be inverted to estimate the latter. This procedure is implemented by means of matrix inversion in the second stage of their original method. However, this procedure requires the inversion of a large matrix, the size of which increases with , and poses computational problems for the actual implementation.
HMSS (1994) proposed an alternative approach to Hotz and Miller’s second step. They suggest running many forward simulations based on the first-stage CCPs. With sufficiently many simulations, the implied value function and its underlying structural parameters can be estimated. This principle underlies AlphaGo’s success in constructing a useful evaluation function for Go.
Although AlphaGo is developed for the game of Go and therefore a dynamic game, its development goal is a “strong” program to beat human champions (and not about studying the strategic interactions among multiple human players in the data). Consequently, AlphaGo’s connection to the empirical dynamic-game methods seems limited. For example, Bajari, Benkard, and Levin (2007, henceforth BBL) extended HMSS (1994) to dynamic games and proposed a moment-inequality-based estimation approach. The development process of game AIs (including AlphaGo) typically abstracts from strategic interactions.181818Section 6 discusses this and other related issues.
Component 4: MCTS and Ensemble
The actual play of AlphaGo is generated by a complex combination of the estimated/reinforced policy function, the value function, and MCTS.
The MCTS part involves randomly playing out many games. This “random” play is generated from a simple version of the empirical policy function that resembles a standard logit form (i.e., one without the multi-layer architecture of DNN to compute as in equation 9). Hence, AlphaGo in the actual game is a hybrid of the following:
the reinforced version of top human players’ strategy (as represented in a deep, convolutional logit functional form),
their implicit value function (with a similar DNN specification), and
real-time forward simulation based on the estimated “quick-and-dirty” policy function (in a simple logit form).
AlphaGo Zero: All-in-One Package
The new version of the program published in 2017, AlphaGo Zero (Silver et al. ), does not use any human data (or handcrafted features to represent the board state). Because it has no empirical (human data-related) component, this new AI is not an obvious subject in terms of econometrics.191919See paragraphs on RL policy function in the earlier part of this section.
Nevertheless, one aspect of its architectural design seems intriguing: a single neural network to perform the functions of both the policy network and value network in the original version. This single network is larger than the previous networks and is combined with MCTS for purely simulation-based search for the optimal strategy to play Go.
This design change is reasonable from the perspective of economic modeling, because the construction of all three of the policy network, value network, and MCTS algorithm in the original AlphaGo was conceptually redundant. Policy and value are dual objects; one implies the other. Likewise, MCTS is a search for the optimal strategy on its own. The “ensemble” effect from combining the three components might have conferred an additional performance gain, but it is ultimately a manifestation of approximation errors within individual components.
Finally, much of the performance gains in AlphaGo Zero seem to stem from the significantly larger size of the neural network architecture (i.e., a more flexible functional form to parameterize the state space). Hence, its superior performance against the original version does not necessarily speak to the costs and benefits of using human data by themselves.
6 Implicit Assumptions
Sections 3, 4, and 5 explained how the concepts and algorithms behind the three game AIs correspond to more familiar ideas and methods in the economics of dynamic structural models.
Deep Blue is a calibration of a linear value function.
Bonanza’s machine-learning method is equivalent to Rust’s (1987) NFXP algorithm.
AlphaGo’s SL policy network is a DNN implementation of Hotz and Miller’s (1993) first-stage nonparametric CCP estimator.
AlphaGo’s SL/RL value network is a DNN implementation of HMSS’s (1994) second-stage CCS estimator.
This section clarifies some of the implicit assumptions underlying (the algorithms to develop) these AIs, and discusses their implications.
Deep Blue does not use data analysis (at least for its main component), but both Bonanza and AlphaGo use logit-style discrete-choice models, and therefore implicitly assume the presence of an error term associated with each of the available actions : type-1 extreme value.202020Because the goal of these games is to win, does not contribute to the eventual payoff . The AIs’ formulation treats as a purely transient, random component of payoffs that players care about only at the time of choosing concurrent .
The inclusion of this continuous random variable to each discrete alternative eliminates the possibility of ties between the payoffs of multiple actionsand , thereby making the mapping between the value function and the policy function unique.
In the plain-vanilla application of discrete-choice models, including Bonanza and AlphaGo, this error term is assumed i.i.d. across actions, players, time, and games. In some empirical contexts, however, one might want to consider relaxing this distributional assumption.
(1) Consideration Sets and Selective Search For example, a subset of legal moves could belong to joseki, or commonly known move patterns, whereas other moves are not even considered by human experts (i.e., outside their “consideration sets”) unless new research shows their effectiveness. Similarly, experts are forward-looking but focus on only a few moves per decision node, conducting a highly selective game-tree search. Capturing these aspects of human play would require the econometrician to distinguish between choice sets and consideration sets.
(2) Cross-sectional Heterogeneity Large , , and necessitate the data-analysis part of the AIs to pool data from all games regardless of the identity of players or occasions. However, players are heterogeneous in their styles and strengths. Such differences would become a source of systematic heterogeneity in across players and contradict the i.i.d. assumption.
(3) Inter-temporal Heterogeneity Likewise, the state of knowledge about desirable moves, or joseki, evolves over time as a result of new games, experimentation, and research (including the emergence of game AIs and their play styles). The evolving nature of knowledge and strategies becomes a source of systematic heterogeneity in across time.
(4) Strategic Interactions Each game in the data embodies two specific players’ attempts to out-maneuver each other. When experts prepare for games, they study rival players’ past and recent strategies to form beliefs about their choice probabilities and exploit any weaknesses. Such interactions continue during the actual games, as players keep updating their beliefs and adjust strategies accordingly. By contrast, both the development process and the actual play of game AIs are based on the assumption that (and more or less equivalently, ), abstracting from strategic interactions.
(5) Time and Physical Constraints The exposition so far has mostly abstracted from the notion of time constraints, but official games impose limits on the amount of time each player can spend on thinking. This time constraint makes the game nonstationary in terms of clock time, and adds another dimension to the player’s optimization problem in terms of the intertemporal allocation of thinking time. The data analysis for the game AIs abstracts from these fundamental aspects of human play, although computers face the same constraint in actual games. For AIs, time constraints manifest themselves either as the length of a truncated game tree (in the case of Deep Blue and Bonanza) or as the number of play-out simulations for MCTS (in the case of AlphaGo).
Similar constraints exist in terms of physical (or mental) capacity in terms of computation speed, information storage, and precision of these operations. Human players are more constrained than computers and more prone to obvious mistakes, especially under severe time constraints. In fact, an important aspect of strategic interactions between human experts is about encouraging the opponent to make mistakes. Mistakes would make the implicit irregular, and explicitly incorporating time and physical constraints would lead to different modeling approaches.
7 Opportunities for Future Research
Relaxing the Implicit Assumptions to Capture Human Behavior
Relaxing these implicit assumptions (in the previous section) would make the underlying models of human behavior for game AIs more realistic (i.e., more human-like). Having achieved performance milestones in terms of pure strength (i.e., approximating the optimal strategy better than top human players), approximating human behavior could be one of the renewed research goals for game AI development. Adding ad hoc features to emulate humans is one way, but developing and estimating a more realistic model could be another, perhaps more fundamental approach.
The econometrics of dynamic structural models have advanced considerably
since the time of Rust (1987), Hotz and Miller (1993), and HMSS (1994) to
address the issues in section 6. Incorporating various kinds of
heterogeneity as well as analyzing strategic interactions has been central
to this progress.212121 Bajari, Benkard, and Levin (2007), Aguirregabiria and Mira (2007), Pakes,
Ostrovsky, and Berry (2007), and Pesendorfer and Schmidt-Dengler (2008)
proposed methods for analyzing dynamic games along the lines of the two-step
estimation method, whereas recent empirical applications, such as Igami
(2017, 2018), Zheng (2016), Yang (2017), and Igami and Uetake (2017), build
on the full-solution method.
Kasahara and Shimotsu (2009) propose a method (based on rank conditions of
the state transition dynamics) to identify the lower bound of the number of
unobserved types that is required to rationalize data patterns. Arcidiacono
and Miller (2011) use an expectation-maximization algorithm to estimate CCPs
in the presence of such unobserved types. Berry and Compiani (2017) advance
an instrumental-variables approach to address unobserved heterogeneity in
Bajari, Benkard, and Levin (2007), Aguirregabiria and Mira (2007), Pakes, Ostrovsky, and Berry (2007), and Pesendorfer and Schmidt-Dengler (2008) proposed methods for analyzing dynamic games along the lines of the two-step estimation method, whereas recent empirical applications, such as Igami (2017, 2018), Zheng (2016), Yang (2017), and Igami and Uetake (2017), build on the full-solution method. Kasahara and Shimotsu (2009) propose a method (based on rank conditions of the state transition dynamics) to identify the lower bound of the number of unobserved types that is required to rationalize data patterns. Arcidiacono and Miller (2011) use an expectation-maximization algorithm to estimate CCPs in the presence of such unobserved types. Berry and Compiani (2017) advance an instrumental-variables approach to address unobserved heterogeneity in dynamic models.Such new methods can be applied to the task of making the AIs’ underlying models more realistic.
Structural Econometrics for “Explainable AI”
Relaxing the implicit econometric assumptions would make the models not only more realistic, but also more interpretable. One of the benefits of developing and estimating a structural model is that the results are economically interpretable, above and beyond the basic notions of causation and correlation in simpler settings (e.g., determining a statistical relationship between some variables and ). The words “interpretable” and “explainable” could mean different things in different fields, but the concept of “structural interpretability” seems useful as a guide for a more formal definition.
Note this proposal about structural interpretation should not be confused with the challenge concerning “explaining DNNs.” DNNs are a flexible functional form, or a class of basis functions for nonparametric estimation, and therefore do not have economic interpretation by themselves. By contrast, the object for which these functional forms are specified could have a structural interpretation (e.g., AlphaGo’s SL policy network is a CCP estimate of the average professional player’s strategy under the maintained assumptions of homogeneity etc.).
DNN for Nonparametric CCP Estimation
The use of DNN specifications for the first-stage nonparametric estimation of CCPs seems a good idea. This class of model specification has long been known to be capable of approximating arbitrary functions, but AlphaGo offers a proof of concept in the dynamic-game context, which is sufficiently complicated and potentially relevant for economic applications.
Clarifying the mapping between the two fields is only a first step toward cross-fertilization, but the opportunities for future research seem to suggest themselves.
Appendix: Functional-Form Specification of AlphaGo
AlphaGo’s policy function uses the following functional form, and its value function has a similar architecture. It consists of 48 input “channels” (variables), 13 “layers” (stages within a hierarchical architecture), and 192 “kernels” (filters to find local patterns). A complete review of deep neural networks in general (or AlphaGo’s model specification in particular) is beyond the scope of this paper, but these objects interact as follows. Each of the 48 channels represents a binary indicator variable that characterizes :
“Features” include the positions of black stones, white stones, and blanks (see Extended Data Table 2 of Silver et al.  for the full list).
These ’s are not combined linearly (as in or ) but are processed by many kernels across multiple hierarchical layers. In the first layer, each of the 192 kernels is a grid with 25 parameters that responds to a particular pattern within 25 adjacent locations.222222The “patterns” that these kernels are designed to pick up should be distinguished from the initial 48 “features” (variables) representing the board state in the original input data. As the name “kernel” suggests, this matrix is applied to perform convolution operations at every one of the locations within the board:
where is the result of convolution for row and column , is the weight for kernel , row , and column , denote the row and column of the kernel, is an input, and is the intercept term (“bias”). The weights and intercepts constitute the parameters of the model (collectively denoted by in the main text). DNNs of this type are called convolutional neural networks (CNNs) in the machine-learning literature, and are primarily used for image-recognition tasks. The results of convolution are subsequently transformed by a function,
whereoutput is smaller than the board; the margins are filled by zeros to preserve the
dimensionality (“zero padding”).
In the second layer, another set of 192 kernels is used to perform convolution on the outputs from the first layer. The results go through the ReLU transformation again, and proceed to the third layer. The size of the kernels in layers 2 through 12 is , instead of in layer 1. In layer 13, the size of the layer is , because the goal of the policy network is to put a number on each of the board positions without “zero padding” the margins. Each of the outputs in this last layer goes through a logit-style monotonic (“softmax”) transformation into the interval,
so that the final output can be interpreted as the players’ conditional choice probabilities of choosing action (or board location ).
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