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[MATH] , reflecting the quality of its fit. On the other hand, it may depart significantly from [MATH] , reflected in a large value of |
[MATH] , the minimum number of editing steps required to create [MATH] if beginning from [MATH] We trade these off in a constraint formulation. Given intended trajectory |
[MATH] and current program [MATH] , we define our synthesis problem as: [EQUATION] In all experiments below, we choose [MATH] . This choice was made to ensure practical run times. |
Algorithms 4.1 IDPS Our first algorithm is a variant of iterative deepening, which we call iterative deepening program search (IDPS). We use a path-checking depth-limited search as a subroutine to minimize space requirements. Unlike traditional AI search procedures, IDPS returns not one but a sequence of programs |
[MATH] whose Hausdorff distance from the target trajectory [MATH] strictly decreases. At iteration [MATH] , we initialize [MATH] to |
[MATH] and expand the search graph up to depth [MATH] . Then, we iterate through all depth- [MATH] programs [MATH] , checking if [MATH] When such a program is found, we emit it to our output sequence, and update |
[MATH] to [MATH] so that future goal states must be strictly better than [MATH] Figure displays an example of such a sequence, where [MATH] is a noisy square and the initial program draws only a line. By convention, we include the initial program [MATH] in the sequence. |
4.2 Sampling search Our second algorithm is a sample-based search shown in Algorithm . Here, we use a corpus of user programs and trajectories to guide our search. Algorithm |
can be described as searching from a graph rooted at [MATH] . The initial program [MATH] is represented by a sequence of editing commands. We use the notation [MATH] to represent the program [MATH] resulting in appending command [MATH] to the end of command list [MATH] . The algorithm samples commands [MATH] from a dis... |
[MATH] parameterized by command sequence [MATH] . A budget of [MATH] candidates are sampled in total. Each of the [MATH] sampling rounds begins at [MATH] |
and samples [MATH] commands, sequentially appending and evaluating them. The candidate minimizing [MATH] among all samples candidates is returned. |
Budget [MATH] , cost [MATH] , initial program [MATH] , visual specification [MATH] Program [MATH] procedure SamplingSearch [MATH] |
[MATH] for each [MATH] in [MATH] do for each [MATH] in [MATH] do [MATH] [MATH] if [MATH] then [MATH] return [MATH] Algorithm 1 Sampling Search |
What distinguishes variants of this algorithm is how the distribution [MATH] is modeled. We factor [MATH] this into a bigram model over command types and a distribution over command arguments. That is, we map the command sequence [MATH] |
to a coarsened sequence [MATH] by discarding arguments so that [MATH] {Get, Remove, Connect, Change, Separate}. We then draw [MATH] from [MATH] , a Markov chain over the coarsened tag sequence. |
To sample [MATH] from [MATH] , we have two models: uniform model and a non-uniform model. On the uniform model, we uniformly sample the [MATH] [MATH] [MATH] |
arguments from the command language from all available types, values, and – in the case [MATH] – all available positions in the current program. |
For the non-uniform model, we make the following observation about the process of editing programs: locations to modify the current program are chosen with a particular focus in mind. Specifically, when a user is connecting a block to another, the source block is more often the last block added to the workspace, while ... |
All of the these probabilities in the above models can be estimated from our corpus. We estimate the transition probabilities [MATH] over the coarsened command sequences smoothing with pseudo-counts of 1. We estimate |
[MATH] and [MATH] by taking the empirical proportion of such decisions over the corpus. The product of the distribution over command types and the distribution over command arguments defines the distribution |
[MATH] shown in Algorithm 4.3 A Computational Speedup While the Hausdorff distance [MATH] , used in both IDPS and sampling search algorithms, is a natural choice for scoring the quality of a fit, the computation of [MATH] becomes prohibitive, as it is quadratic in the number of points of the two point sets [MATH] and [... |
Evaluation We solicit a corpus of [MATH] programs and their visual specifications from 11 volunteer study participants, who range in programming experience from novice to professional. Figure |
and Figure give an examples of a participant program and some participant specifications, respectively. To construct this corpus, we employ the following data collection procedure: |
Step 1 Each participant is educated in the capabilities of our turtle language and the editor environment by completing an introductory set of exercises. |
Step 2 The participant is instructed to draw a trajectory on a standard canvas. Let [MATH] represent this visual specification of the intended of the program, as drawn by participant [MATH] |
Step 3 The participant is instructed to compose a program in the turtle language which follows the drawn trajectory as closely as possible. Let [MATH] represent the matching program from participant [MATH] |
We record each step in the participant’s programming process as a formal editing command, c.f. Section 2.2 . We represent the complete program |
[MATH] as the sequence of editing commands [MATH] which produce [MATH] if performed in the editor begining at an empty workspace. |
Now consider each of these programs with the final [MATH] commands removed, that is [MATH] . Letting [MATH] denote the search algorithm at hand, our interest is in evaluating the performance of [MATH] , the [MATH] -ahead performance of our search algorithm. |
As our search procedure does not discriminate between syntactically differing programs which produce the same trajectory (share the same semantics), we consider programs up to semantic equivalence. We define the semantic equivalence class of [MATH] as |
[MATH] . We then define our [MATH] -ahead accuracy in terms of our procedure achieving any semantically equivalent program to the target. |
Unfortunately, if we only consider [MATH] correct when [MATH] , we overlook the important case where [MATH] is a better fit for [MATH] than [MATH] . This may happen, for example, if the participant has little programming experience and writes [MATH] incorrectly. A softer measure of accuracy is simply the Hausdorff dist... |
to [MATH] Table summarizes these [MATH] -ahead metrics. We evaluate our method against all metrics. We also report runtime. Related Work |
The problem of generating a computer program from some specification has been studied since the beginnings of AI. Relevant work here falls into the two broad camps of synthesis where an explicit program is generated and |
induction where a latent representation may be used to generate input output pairs In the inductive setting, there is a large literature of work relating to the search for algorithms which correct a sketch. |
uses recurrent neural networks to model conditional sketch generation, an image-to-image transformation problem. uses probabilistic program induction to perform one-shot modeling. In the programming synthesis community, there is a large body of work synthesizing program from logical specifications |
Synthesis settings can further be distinguished by whether a given specification is partial or total . Abstractly, we may think of synthesis as attempting to infer some [MATH] where [MATH] is a family of programs taking inputs from [MATH] and outputting [MATH] . A partial specification comes in the form of a set of inp... |
[MATH] whose input items are a proper subset of [MATH] . This induces a problem of inference as the synthesis algorithm must settle on some choice of output for unobserved input. By way of contrast, a total specification fully specifies [MATH] as a function. But even if we know the specification, we may have difficulty... |
A further distinction can be made between settings where the specification is noisy or noiseless . More recent work such as and naturally handle noisy specifications owing to their use of neural models. |
and allow the user to sketch a partial program in addition to a specification through inputs/output pairs. Our method is distinct in that the user provides not a sketch but a partial program which may contain errors. That is, our synthesis is not constrained by the given partial program. |
Results We compute [MATH] -ahead completions for [MATH] . To ensure practical run times, we allot a state budget of [MATH] programs for each algorithm and [MATH] . Likewise, we enforce a static horizon of [MATH] so that [MATH] |
for all completions. Figure (a) plots mean [MATH] for each algorithm against [MATH] . For small [MATH] such as [MATH] , IDPS will always recover [MATH] unless a better fit is nearby, while sampling search manages a less reliable 63% recovery rate. Under our state budget constraint, IDPS’s performance decreases almost m... |
(b) and Figure (c), which plot mean [MATH] and mean [MATH] respectively against [MATH] , reflect that sampling search has greater opportunity to explore deeper and more structured programs. |
The dashed line of Figure (b) represents the mean Hausdorff distances of the user’s true completion [MATH] , that is, the mean of |
[MATH] across our corpus. As we regard the trajectory [MATH] as the true label, we can see that for smaller values of [MATH] our algorithms improved upon the user’s completed program, [MATH] . For small lookaheads, this suggests that users may benefit from viewing programs returned by our synthesis method. |
Nonuniform sampling search dominates the sampling search regime, outperforming uniform sampling in [MATH] and [MATH] for all [MATH] . By localizing block targets in a manner statistically more consistent with observed human behavior, the algorithm restricts its search to programs produced by high-level groupings of com... |
It is worth noting that both uniform and nonuniform sampling search converge on better-fit trajectories when faced with idiosyncratic programming techniques by the programmer. For example, one participant wrote a large loop body and only added the loop itself as his last step, while most other programmers added the loo... |
Figure shows that sampling search is consistently faster than IDPS, the nonuniform (resp. uniform) variant requiring an average of 5.97 minutes (resp. 4.82 minutes) for convergence as opposed to 9.83 minutes. That the former is nearly twice as fast as the latter is not surprising. For both the uniform and nonuniform va... |
A qualitative look at some specific results can further insight. For example, consider item 18 from our corpus as presented in Figure |
Here we show in the top row the solution obtained by the uniform algorithm for lookaheads [MATH] and [MATH] and in the bottom row the user-drawn trajectory and the nonuniform solution, which was the same in both cases of [MATH] . As we can see the uniform algorithm returns a solution which “cheats” with respect to our ... |
repeat s tend to draw such trajectories. The nonuniform solution, modeling as it does the attention or focus of the programmer, overcame these shortcomings. |
Nevertheless, the nonuniform algorithm did not perform strictly better. For example, item 3 in our corpus is presented in Figure |
10 . Here, the uniform algorithm returned a reasonable solution, while the nonuniform seemed to lose its way. This example is notable for another reason in that it demonstrates how the Hausdorff distance may not encode all features of the user’s intention. We see here that the program returned by the |
uniform algorithm adds a bend to the line inside the square, while the user’s trajectory is suggests a straight curve. This raises the following question: is the angle of the line within the square the more salient feature of the user’s intention or its straightness? The Hausdorff distances “fudges” by adding a curve, ... |
As the quantitative results show, however, the nonuniform algorithm fared significantly better. We wish to emphasize the conceptual significance of this better performance. From a statistical point of view, we would argue that our corpus is not drawn from the “true distribution” for our task, namely some kind of distri... |
Conclusion We have formulated program synthesis with visual specification in the frame of classical AI search and have proposed two algorithms. Sampling methods produce improved solutions and scale more readily to larger problem instances. A sampling method informed by the attention and distribution of behaviors observ... |
Acknowledgments We would like to thank Scott Alfeld for insightful comments and discussion. This work is supported by NSF grant 1423237. |
# Source: arxiv 1806.00955 # Title: A Game-Theoretic Approach to Recommendation Systems with Strategic Content Providers # Sections: all # Downloaded: 2026-03-02T09:21:58.868051+00:00 |
A Game-Theoretic Approach to Recommendation Systems with Strategic Content Providers Abstract We introduce a game-theoretic approach to the study of recommendation systems with strategic content providers. Such systems should be fair and stable. Showing that traditional approaches fail to satisfy these requirements, we... |
Introduction Recommendation systems (RSs hereinafter) have rapidly developed over the past decade. By predicting a user preference for an item, RSs have been successfully applied in a variety of applications. Moreover, the amazing RSs offered by giant e-tailers and e-marketing platforms, such as Amazon and Google, lie ... |
. Consider the following cases: [MATH] When Alice drives her car, her personal assistant runs the default navigation application. When she makes a stop at a junction, the personal assistant may show Alice advertisements provided by neighborhood stores, or an update on the stock market status as provided by financial br... |
[MATH] Bob is reading news of the day on his mobile application. The application, aware of Bob’s interests, is presenting news deemed most relevant to him. The news is augmented by advertisements, provided by competing content providers, as well as articles by independent reporters. The mobile application, balancing Bo... |
In these contexts, the RS integrates information from various providers, often sponsored content, which is probably relevant to the user. The content providers are strategic –- namely, make decisions based on the way the RS operates, aiming at maximizing their exposure. For instance, to draw Bob’s attention, a content ... |
In this paper, we study ways of overcoming this dilemma using canonical concepts in game theory to impose two requirements on the RS: fairness and stability. Fairness is formalized as the requirement of satisfying fairness-related properties, and stability is defined as the existence of a pure Nash equilibrium. Analyzi... |
Our first result is that traditional RSs fail to satisfy both of the above requirements. Traditional RSs are complete, in the sense that they always show some content to the user, but it turns out that this completeness property cannot be satisfied simultaneously with the fairness and equilibrium existence requirements... |
, which is a celebrated mechanism for value distribution in game-theoretic contexts (see, e.g., ). In our work, it is proposed as a tool for recommendations, namely for setting display probabilities. Since the Shapley value is employed in countless settings for fair allocation, it is not surprising that it satisfies ou... |
, and therefore any better-response learning dynamics converge to an equilibrium (see, e.g., ). Note that this far exceeds our minimal stability requirement from the RS. |
Implementation in commercial products would require the mediator to be computationally tractable. The mediator interacts with users; hence a fast response is of great importance. In another major result, we show that the Shapley mediator has a computationally efficient implementation. The latter is in contrast to the i... |
Another essential property of the Shapley mediator is economic efficiency . Unlike cooperative games, where the Shapley value can be characterized as the only solution concept to satisfy properties equivalent to fairness and economic efficiency, in our setting the Shapley mediator is not characterized solely by fairnes... |
1.1 Related work This work contributes to three interacting topics: fairness in general machine learning, multi-stakeholder RSs and game theory. |
The topic of fairness is receiving increasing attention in machine learning and data mining . A major line of research is discrimination aware classification |
where classification algorithms must maintain high predictive accuracy without discriminating on the basis of a variable representing membership in a protected class, e.g. ethnicity. In the context of RSs, the work of Kamishima et al. |
addresses a different aspect of fairness (or lack thereof): bias towards popular items. The authors propose a collaborative filtering model which takes into account viewpoints given by users, thereby tackling the tendency for popular items to be recommended more frequently, a problem posed in |
A related problem is over-specialization, i.e., the tendency to recommend items similar to those already purchased or liked in the past, which is addressed in |
Zheng surveys multi-stakeholder RSs, and highlights practical applications. Examples include RSs for sharing economies (e.g. AirBnB, Uber, etc.), online dating |
, and recruiting Burke discusses fairness in multi-stakeholder RSs, and presents a taxonomy of classes of fairness-aware RSs. The author distinguishes between user fairness, content provider fairness and pairwise fairness, and reviews applications for these fairness types. A practical problem concerning fairness in mul... |
. In their work, an online platform is used by users who play two roles: customers seeking recommendations and content providers aiming for exposure. They report, based on empirical evidence, that collaborative filtering techniques tend to create rich-gets-richer scenarios, and propose a method for re-ranking scores, i... |
Note that all the work above considers traditional machine learning tasks that enforce upon the solution some form of fairness, as defined specifically for each task. They suggest additional considerations, but do not consider that the parties (i.e., users, content providers) will change their behavior as a result of t... |
Finally, strategic aspects of classical machine learning tasks were also introduced recently . The idea that a recommendation algorithm affects content-provider policy, and as a result must be accompanied by a game-theoretic study is key to recent works in search/information retrieval |
; so far, however, such work has not dealt with the issue of fairness. Problem formulation From here on, our ideas will be exemplified in the following motivational example: a mobile application (or simply app) is providing users with valuable content. A set of players (advertisers) publish their items (advertisements)... |
Formally, the recommendation game is defined as follows: A set of users [MATH] , a set of players [MATH] , and a mediator [MATH] |
The set of items (e.g. possible ad formats/messages to select from) available to player [MATH] is denoted by [MATH] , which we assume to be finite. A strategy of player [MATH] is an item from [MATH] |
Each user [MATH] has a satisfaction function [MATH] , where [MATH] is the set of all available items. In general, [MATH] measures the satisfaction level of [MATH] w.r.t. [MATH] |
When triggered by the app, [MATH] decides which item to display, if any. Formally, given the strategy profile [MATH] and a user [MATH] [MATH] is a distribution over [MATH] , where [MATH] symbolizes maintaining the plain content of the app. That is, displaying no item at all. We refer to [MATH] |
as the probability that player [MATH] ’s item will be displayed to [MATH] under the strategy profile [MATH] Each player gets one monetary unit when her item is displayed to a user. Therefore, the expected payoff of player [MATH] under the strategy profile [MATH] is [MATH] |
The social welfare of the players under the strategy profile [MATH] is the expected number of displays, [MATH] For ease of notation, we shall sometimes refer to [MATH] as the maximum satisfaction level of user [MATH] from the items in [MATH] , i.e., [MATH] |
We demonstrate our setting with the following example. Example 1 Consider a game with two players and three users. Let [MATH] such that the satisfaction levels of the users with respect to the items are |
[EQUATION] Consider a mediator displaying each user with the most satisfying item to her taste, denoted by TOP . For example, [MATH] , since [MATH] The profile [MATH] will probably be materialized in realistic scenarios, since the payoff of player 1 under the strategy profile [MATH] is [MATH] , while [MATH] . Notice th... |
[MATH] , while [MATH] ; hence, the users suffer from strategic behavior of the players. After defining general recommendation games, we now present a few properties that one may desire from a mediator. First and foremost, a mediator has to be fair . The following is a minimal set of fairness properties: |
Null Player . If [MATH] , then it holds that [MATH] . Informally, an item will not be displayed to [MATH] if it has zero satisfaction level w.r.t. him. |
Symmetry . If [MATH] has the same satisfaction level from two items, they will be displayed with the same probability. Put differently, if [MATH] , then [MATH] |
User-Independence . Given the selected items, the display probabilities depend only on the user: if user [MATH] is removed from/added to [MATH] [MATH] will not change, i.e., |
[EQUATION] Leader Monotonicity [MATH] displays the most satisfying items (w.r.t. a specific user) with higher probability than it displays other items. Formally, if [MATH] and [MATH] , then [MATH] |
For brevity, we denote the above set of fairness properties by . In addition, an essential property in a system with self-motivated participants is that it will be stable. Instability in such systems is a result of a player aiming to improve her payoff given the items selected by others. A minimal requirement in this r... |
Stability Under any set of players, available items, users and user satisfaction functions, the game induced by [MATH] possesses a PNE. |
For brevity, we denote this property by . The goal of this paper is to devise a computationally tractable mediator that satisfies both and |
2.1 Impossibility of classical approaches We highlight a few benchmark mediators in Table , including TOP , which was introduced informally in Example . Another interesting mediator is BTL , which follows the lines of the Bradley-Terry-Luce model |
BTL is addressed here as a representative of a wide family of weight-based mediators: mediators that distribute display probability according to weights, determined by a monotonically increasing function of the user satisfaction (e.g., softmax). Common to [MATH] and any other weight-based mediator, is that an item is d... |
Complete . For any recommendation game and any strategy profile [MATH] [MATH] Since the goal of an RS is to provide useful content to users, satisfying Complete seems justified. Although it seems unreasonable to avoid showing any content to a certain user at a certain time, it turns out that this avoidance is crucial i... |
Theorem 1 No mediator can satisfy [MATH] and Complete Proof sketch. We construct a game with two players, three users and three strategies, and show that no mediator can satisfy [MATH] and Complete . Importantly, our technique can be used to show that any arbitrary game does not possess a PNE or that a slight modificat... |
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