Present-Biased Optimization
PPresent-Biased Optimization ∗ Fedor V. Fomin † Pierre Fraigniaud ‡ Petr A. Golovach † January 1, 2021
Abstract
This paper explores the behavior of present-biased agents, that is, agents who erroneouslyanticipate the costs of future actions compared to their real costs. Specifically, the paperextends the original framework proposed by Akerlof (1991) for studying various aspects ofhuman behavior related to time-inconsistent planning, including procrastination, and aban-donment, as well as the elegant graph-theoretic model encapsulating this framework recentlyproposed by Kleinberg and Oren (2014). The benefit of this extension is twofold. First, itenables to perform fine grained analysis of the behavior of present-biased agents dependingon the optimisation task they have to perform. In particular, we study covering tasks vs.hitting tasks, and show that the ratio between the cost of the solutions computed by present-biased agents and the cost of the optimal solutions may differ significantly depending on theproblem constraints. Second, our extension enables to study not only underestimation offuture costs, coupled with minimization problems, but also all combinations of minimiza-tion/maximization, and underestimation/overestimation. We study the four scenarios, andwe establish upper bounds on the cost ratio for three of them (the cost ratio for the originalscenario was known to be unbounded), providing a complete global picture of the behaviorof present-biased agents, as far as optimisation tasks are concerned.
Present bias is the term used in behavioral economics to describe the gap between the anticipatedcosts of future actions and their real costs. A simple mathematical model of present bias wassuggested by Akerlof [1]. In this model the cost of an action that will be perceived in the futureis assumed to be β times smaller than its actual cost, for some constant β <
1, called the degreeof present bias . The model was used for studying various aspects of human behavior related to time-inconsistent planning, including procrastination , and abandonment .Kleinberg and Oren [10, 11] introduced an elegant graph-theoretic model encapsulatingAkerlof’s model. The approach of Kleinberg and Oren is based on analyzing how an agentnavigates from a source s to a target t in a directed edge-weighted graph G , called task graph .At any step, the agent chooses the next edge to traverse from the current vertex v thanks toan estimation of the length of the shortest path from v to t passing through each edge outgoingfrom v . A crucial characteristic of the model is that the estimation of the path lengths is present-biased . More specifically, the model of Kleinberg and Oren includes a positive parameter β <
1, the degree of present bias, and the length of a path x , . . . , x k from x = v to x k = t in G is evaluated as ω + β (cid:80) k − i =1 ω i where ω i denotes the weight of edge ( x i , x i +1 ), for every i ∈ { , . . . , k − } . As a result, the agent may choose an outgoing edge that is not on any shortest ∗ A preliminary version of this paper is accepted for AAAI 2021. The research received funding from theResearch Council of Norway via the project “MULTIVAL” (grant no. 263317) and from the French NationalResearch Agency (ANR) via the project “DESCARTES”. † Department of Informatics, University of Bergen, Norway ‡ IRIF, Universit´e de Paris and CNRS, France a r X i v : . [ m a t h . O C ] D ec ath from v to t , modeling procrastination by underestimating the cost of future actions to beperformed whenever acting now in some given way. With this effect cumulating along its wayfrom s to t , the agent may significantly diverge from shortest s - t paths, which demonstratesthe negative impact of procrastination. Moreover, the cost ratio , which is the ratio betweenthe cost of the path traversed by an agent with present bias and the cost of a shortest path,could be arbitrarily large. An illustrating example is depicted on Fig. 1, borrowed from [11],and originally due to Akerlof [1]. Among many results, Kleinberg and Oren showed that anygraph in which a present-biased agent incurs significantly more cost than an optimal agent mustcontain a large specific structure as a minor. This structure, called procrastination structure , isspecifically the one depicted on Fig. 1. v1 ts v2 cc c v3 v4 v5c c cx x x x x (a) The Akerlof example sv01v02 v10v11v12 v20v21v22 v30v31t (b)
Homework deadlines
Figure 2: Path problems that exhibit procrastination, abandonment, and choice reduction.graph, it can be crucial to present the agent with a subgraph that includes not just P but also certainadditional nodes and edges that do not belong to P . We give a graph-theoretic characterization ofthe possible subgraphs supporting e cient traversal. Finally, for heterogeneous agents, we explorea simple variant of the problem based on partitioning large tasks into smaller ones.Before turning to these questions, we first discuss the basic graph-theoretic problem in moredetail, showing how instances of this problem capture the time-inconsistency phenomena discussedearlier in this section. In order to argue that our graph-theoretic model captures a variety of phenomena that have beenstudied in connection with time-inconsistency, we present a sequence of examples to illustrate someof the di↵erent behaviors that the model exhibits. We note that the example in Figure 1 alreadyillustrates two simple points: that the path chosen by the agent can be sub-optimal; and that evenif the agent traverses an edge e with the intention of following a path P that begins with e , it mayend up following a di↵erent path P that also begins with e .For an edge e in G , let c ( e ) denote the cost of e ; and for a path P in G , let e i ( P ) denote the i th edge on P . In terms of this notation, the agent’s decision is easy to specify: when standing at anode v , it chooses the path P that minimizes c ( e ( P )) + P i> c ( e i ( P )) over all P that run from v to t . It follows the first edge of P to a new node w , and then performs this computation again.We begin by observing that Figure 2(a) represents a version of the Akerlof example from theintroduction. (For simplicity we assume that the delivery of the package is instantaneous, so h = 0.Also recall that we use b to denote .) Node t represents the state in which the agent has sentthe package, and node v i represents the state in which the agent has reached day i without sendingthe package. The agent has the option of going directly from node s to node t , and this is theshortest s - t path. But if ( b c > bx , then the agent will instead go from s to v , intending tocomplete the path s - v - t in the next time step. At v , however, the agent decides to go to v ,intending to complete the path v - v - t in the next time step. This process continues: the agent,following exactly the reasoning in the example from the introduction, is procrastinating and notgoing to t , and in the end its path goes all the way to the last node v n ( n = 5 in the figure) beforefinally taking an edge to t . (One minor change from the set-up in the introduction is the fact thatthe present-bias e↵ect here holds more consistently, and is applied to x as well; this has no reale↵ect on the underlying story.) Extending the model to include rewards.
Thus far we can’t talk about an agent who5Figure 1:
Procrastination structure as displayed in [11]; Assuming x + β c < c , the path followedby the agent is s , v , . . . , v , t ; The ratio between the length of the path followed by the agentand the shortest s - t path can be made arbitrarily large by adding more nodes v k with k ≥ . In this paper, we are interested in understanding what kind of tasks performed by the agentresult in large cost ratio. Let us take the concrete example of an agent willing to acquire theknowledge of a set of scientific concepts, by reading books. Each book covers a certain numberof these concepts, and the agent’s objective is to read as few books as possible. More generally,each book could also be weighted according to, say, its accessibility to a general reader, or itslength. The agent’s objective is then to read a collection of books with minimum total weight.Both the weight and the collection of concepts covered by each book are known to the agenta priori. This scenario is obviously an instance of the (weighted) set-cover problem. Let usassume, for simplicity, that the agent has access to a time-biased oracle providing it with thefollowing information. Given the subset of concepts already acquired by the agent when itqueries the oracle, the latter returns to the agent a set { b , . . . , b k − } of books minimizing ω + β (cid:80) k − i =1 ω i among all sets of books covering the concepts yet to be acquired by the agent, where ω ≤ ω ≤ · · · ≤ ω k − are the respective weights of the books b , . . . , b k − . This correspondsto the procrastination scenario in which the agent picks the easiest book to read now, andunderestimates the cost of reading the remaining books later. Then the agent moves on byreading b , and querying the oracle for figuring out the next book to read for covering theremaining uncovered concepts after having read book b .The question is: by how much the agent eventually diverges from the optimal set of booksto be read? This set-cover example fits with the framework of Kleinberg and Oren, by definingthe vertex set of the task graph as the set of all subsets of concepts, and placing an edge ( u, v )of weight ω from u to v if there exists a book b of weight ω such that v is the union of u andthe concepts covered by b . In this setting, the agent needs to move from the source s = ∅ tothe target t corresponding to the set of all the concepts to be acquired by the agent. Under thissetting, the question can be reformulated as: under which circumstances the set-cover problemyields a large cost ratio?More generally, let us consider a minimization problem where, for every feasible solution S of every instance of the problem, the cost c ( S ) of S can be expressed as c ( S ) = (cid:80) x ∈ S ω ( x )for some weight function ω . This includes, e.g., set-cover, min-cut, minimum dominating set,2eedback-vertex set, etc. We then define the biased cost c β as c β ( S ) = ω ( x ∗ ) + β c ( S (cid:114) { x ∗ } ) , (1)where x ∗ = arg min x ∈ S ω ( x ). Given an instance I of the minimization problem at hand, theagent aims at finding a feasible solution S ∈ I minimizing c ( S ). It does so using the followingpresent-biased planning, where I = I . Minimization scenario:
For k ≥
0, given an instance I k , the agent computes the fea-sible solution S k with minimum cost c β ( S k ) among all feasible solutions for I k . Let x ∗ k =arg min x ∈ S k w ( x ). The agent stops whenever { x ∗ , x ∗ , . . . , x ∗ k } is a feasible solution for I . Oth-erwise, the agent moves to I k +1 = I k (cid:114) x ∗ k , that is, the instance obtained from I k when oneassumes x ∗ k selected in the solution.This general scenario is indeed captured by the Kleinberg and Oren model, by defining thevertex set of the graph task graph as the set of all “sub-instances” of the instance I at hand,and placing an edge ( u, v ) of weight w from u to v if there exists an element x of weight ω suchthat v results from u by adding x to the current solution. The issue is to analyze how far thesolution computed by the present-biased agent is from the optimal solution. The first questionaddressed in this paper is therefore the following. Question 1.
For which minimization tasks a large cost ratio may appear?
In the models of Akerlof [1] and Kleinberg and Oren [11] the degree β of present bias isassumed to be less than one. However, there are natural situations where underestimating thefuture costs does not hold. For example, in their influential paper, Loewenstein, O’Donoghue,and Rabin [14] gave a number of examples from a variety of domains demonstrating the preva-lence of projection bias . In particular, they reported an experiment by Jepson, Loewenstein,and Ubel [9] who “asked people waiting for a kidney transplant to predict what their qualityof life would be one year later if they did or did not receive a transplant, and then asked thosesame people one year later to report their quality of life. Patients who received transplantspredicted a higher quality of life than they ended up reporting, and those who did not predicteda lower quality of life than they ended up reporting”. In other words, there are situations inwhich people may also overestimate the future costs. In the model of Kleinberg and Oren [11]overestimation bias corresponds to the situation of putting the degree of present bias β > Question 2.
Could a large cost ratio appear for minimization problems when the degree ofpresent bias β is more than ? Reformulating the analysis of procrastination, as stated in Question 1, provides inspirationfor tackling related problems. As a matter of fact, under the framework of Kleinberg andOren, procrastination is a priori associated to minimization problems. We also investigate maximization problems, in which a present-biased agent aims at, say, maximizing its revenueby making a sequence of actions, each providing some immediate gain that the agent maximizeswhile underestimating the incomes resulting from future actions. As a concrete example, letus consider an instance of Knapsack. The agent constructs a solution gradually by pickingthe item x of highest value ω ( x ) in a feasible set { x , . . . , x k − } of items that is maximizing ω ( x ) + β (cid:80) k − i =1 ω ( x i ) for the current sub-instance of Knapsack. In general, given an instance I of a maximisation problem, we assume that the agent applies the following present-biasedplanning, with I = I : 3 aximization scenario: Given an instance I k for k ≥
0, the agent computes the feasiblesolution S k with maximum cost c β ( S k ) among all feasible solutions for I k — where the definitionof x ∗ in Eq. (1) is replaced by x ∗ = arg max x ∈ S w ( x ). With x ∗ k = arg max x ∈ S k w ( x ), the agentstops whenever { x ∗ , x ∗ , . . . , x ∗ k } is an inclusion-wise maximal feasible solution for I , and movesto I k +1 = I k (cid:114) x ∗ k otherwise.We are interested in analyzing how far the solution computed by the present-biased agent isfrom the optimal solution. More generally even, we aim at revisiting time-inconsistent planningby considering both cases β < β >
1, that is, not only scenarios in which the agentunderestimates the cost of future actions, but also scenarios in which the agent overestimates the cost of future actions. The last, more general question addressed in this paper is thereforethe following.
Question 3.
For which optimization tasks, and for which time-inconsistency planning (under-estimation, or overestimation of the future actions), the solutions computed by a present-biasedagent are far from optimal, and for which they are close?
For all these problems, we study the cost ratio (cid:37) = c ( S ) opt (resp., (cid:37) = opt c ( S ) ) where S is thesolution returned by the present-biased agent, and opt = c ( S opt ) is the cost of an optimalsolution for the same instance of the considered minimization (resp., maximization) problem. Focussing on agents aiming at solving tasks, and not just on agents aiming at reaching targetsin abstract graphs, as in the generic model in [11], allows us not only to refine the worst-caseanalysis of present-biased agents, but also to extend this analysis to scenarios corresponding tooverestimating the future costs to be incurred by the agents (by setting the degree β of presentbias larger than 1), and to maximisation problems. Minimization & underestimation.
In the original setting of minimization problems, withunderestimation of future costs (i.e., β < (cid:37) of an agent per-forming k steps, that is, computes a feasible solution { x ∗ , . . . , x ∗ k } , satisfies (cid:37) ≤ k . This is incontrast to the general model in [11], in which an agent can incur a cost ratio exponential in k when returning a k -edge path from the source to the target. Hence, in particular, our mini-mization scenarios do not produce the worst cases examples constructed in [11], i.e., obtainedby considering travels from sources to targets in arbitrary weighted graphs.On the other hand, we also show that a “minor structure” bearing similarities with theone identified in [11] can be identified. Namely, if an agent incurs a large cost ratio, then theminimization problem addressed by the agent includes a large instance of a specific form ofminimization problem. Min/maximization & under/overestimation.
Interestingly, the original setting of mini-mization problems, with underestimation of future costs, is far from reflecting the whole natureof the behavior of present-biased agents. Indeed, while minimization problems with underes-timation of future costs may result in unbounded cost ratios, the worst-case cost ratios corre-sponding to the three other settings can be upper bounded, some by a constant independent ofthe task at hand. Specifically, we show that: • For any minimization problem with β >
1, the cost ratio is at most β ; • For any maximization problem with β <
1, the cost ratio is at most β ;4 For any maximization problem with β >
1, the cost ratio is at most β c , where c ≤ opt isthe cost of a solution constructed by the agent.Our results are summarized in Table 1.minimization maximization β < ∞ [11] 1 /β [Thm 5(i)] β > β [Thm 4] (1 + log β ) opt log opt [Cor 1]Table 1: Upper bounds on the worst case ratio between the solution cost returned by thepresent-biased agent and the optimal solution opt . The symbol ∞ means that the cost ratiocan be arbitrarily large, independently of the values of β , and opt . Let us remark that, for minimization problems with β >
1, as well as for maximizationproblems with β <
1, we have that the cost ratio is bounded by a constant. However, formaximization problems with β >
1, the cost ratio can be exponential in the cost of the computedsolution. We show that this exponential upper bound is essentially tight.
Approximated evaluations.
Actually, in many settings, discrete optimization problems arehard. Therefore, for evaluating the best feasible solution according to the biased cost function c β , an agent may have to solve computationally intractable problems. Thus, in a more realisticscenario, we assume that, instead of computing an optimal solution for c β at every step, theagent computes an α -approximate solution. Fine-grained analysis.
In contrast to the general model in [11], the refined model of thispaper enables fine-grain analysis of the agents’ strategies, that is, it enables identifying differentbehaviors of the agents as a function of the considered optimisation problems. Specifically,there are natural minimization problems for which specific bounds on the cost ratio can beestablished.To illustrate the interest of focusing on optimisation tasks, we study two tasks in detail,namely set-cover and hitting set , and show that they appear to behave quite differently. Forset-cover, we show that the cost ratio is at most d · opt , where d is the maximum size of thesets. For hitting set, we show that the cost ratio is at most d ! ( β opt ) d , again for d equal to themaximum size of the sets. This latter result is using the sunflower lemma of Erd˝os and Rado [6].Finally, we identify a simple restriction of the agent’s strategy, which guarantees that thecost of the solution computed by the agent is not more than β times the cost of an optimalsolution. Our work is directly inspired by the aforementioned contribution of Kleinberg and Oren [10],which was itself motivated by the earlier work by Akerlof [1]. We refer to [10, 11] for a survey ofearlier work on time-inconsistent planning, with connections to procrastination, abandonment,and choice reduction. Hereafter, we discuss solely [10], and the subsequent work. Using theirgraph-theoretic framework, Kleinberg and Oren reasoned about time-inconsistency effects. Inparticular, they provided a characterization of the graphs yielding the worst-case cost-ratio, andthey showed that, despite the fact that the degree β of present bias can take all possible valuesin [0 , s - t paths computed bypresent-biased agents for all degrees of present bias is of size at most polynomial in the numberof nodes. They also showed how to improve the behavior of present-biased agents by deleting5dges and nodes, and they provided a characterization of the subgraphs supporting efficientagent’s behavior. Finally, they analyzed the case of a collection of agents with different degreesof present bias, and showed how to divide the global task to be performed by the agents into“easier” sub-tasks, so that each agent performs efficiently her sub-tasks.As far as we are aware of, all contributions subsequent to [10], and related to our paper,essentially remain within the same graph theoretic framework as [10], and focus on algorithmicproblems related to this framework. In particular, Albers and Kraft [4] studied the ability toplace rewards at nodes for motivating and guiding the agent. They show hardness and in-aproximability results, and provide an approximation algorithm whose performances match theinaproximability bound. The same authors considered another approach in [2] for overcomingthese hardness issues, by allowing not to remove edges but to increase their weight. They wereable to design a 2-approximation algorithm in this context. Tang et al. [15] also proved hardnessresults related to the placement of rewards, and showed that finding a motivating subgraph isNP-hard. Gravin et al. [7] (see [8] for the full paper) extended the model by considering thecase where the degree of present bias may vary over time, drawn independently at each stepfrom a fixed distribution. In particular, they described the structure of the worst-case graph forany distribution, and derived conditions on this distribution under which the worst-case costratio is exponential or constant.Kleinberg, Oren, and Raghavan [12, 13] revisited the model in [10]. In [12], they wereconsidering agents estimating erroneously the degree β of present bias, either underestimatingor overestimating that degree, and compared the behavior of such agents with the behavior of“sophisticated” agents who are aware of their present-biased behavior in future and take thisinto account in their strategies. In [13], they extended the model by considering not only agentssuffering from present-biases, but also from sunk-cost bias, i.e., the tendency to incorporatecosts experienced in the past into one’s plans for the future. Albers and Kraft [3] considered amodel with uncertainty, bearing similarities with [12], in which the agent is solely aware thatthe degree of present bias belongs to some set B ⊆ (0 , This section includes a formal definition of inconsistent planning by present-biased agents, anddescribes two extreme scenarios: one in which a present-biased agent constructs worst caseplannings, and one in which the plannings generated by a present-biased agent are close tooptimal.
We consider minimization problems defined as triples ( I , F, c ), where I is the set of instances(e.g., the set of all graphs), F is a function that returns the set F ( I ) of feasible solutions forevery instance I ∈ I (e.g., the set of all edge-cuts of any given graph), and c is a non-negativefunction returning the cost c ( I, S ) of every feasible solution S ∈ F ( I ) of every instance I ∈ I (e.g., the number of edges in a cut). We focus solely on optimization problems for which(i) a finite ground set S I (cid:54) = ∅ is associated to every instance I ,(ii) every feasible solution for I is a set S ⊆ S I , and(iii) c ( I, S ) = (cid:80) x ∈ S ω ( x ) where ω : S I → N is a weight function.Moreover, we enforce two properties that are satisfied by classical minimization problems.Specifically we assume that: 6 All considered problems are closed downward, that is, for every considered minimizationproblem ( I , F, c ), every I ∈ I , and every x ∈ S I , the instance I (cid:114) { x } defined by thefeasible solutions S (cid:114) { x } , for every S ∈ F ( I ), is in I with the same weight function ω asfor I . This guarantees that an agent cannot be stuck after having performed some task x ,as the sub-problem I (cid:114) { x } remains solvable for every x . • All considered feasible solutions are closed upward, that is, for every minimization problem( I , F, c ), and every I ∈ I , S I is a feasible solution, and, for every S ∈ F ( I ), if S ⊆ S (cid:48) ⊆ S I then S (cid:48) ∈ F ( I ). This guarantees that an agent performing a sequence of tasks x , x , . . . eventually computes a feasible solution.Inconsistent planning can be rephrased in this framework as follows. Inconsistent Planning.
Let β < I , F, c ), the biased cost c β satisfies c β ( S ) = ω ( x ) + β c ( S (cid:114) { x } ) for every feasible solution S ofevery instance I ∈ I , where x = arg min y ∈ S ω ( y ). Given an instance I , the agent aims at findinga feasible solution S ∈ I by applying a present-based planning defined inductively as follows.Let I = I . For every k ≥
0, given the instance I k , the agent computes a feasible solution S k with minimum cost c β ( S k ) among all feasible solutions for I k . Let x k = arg min y ∈ S k ω ( y ). Theagent stops whenever { x , x , . . . , x k } is a feasible solution for I . Otherwise, it carries on theconstruction of the solution by considering I k +1 = I k (cid:114) { x k } .Observe that inconsistent planning terminates. Indeed, since the instances of the consideredproblem ( I , F, c ) are closed downwards, I k = I (cid:114) { x , . . . , x k − } ∈ I for every k ≥
0, i.e.,inconsistent planning is well defined. Moreover, since the feasible solutions are closed upward,there exists k ≥ { x , x , . . . , x k } is a feasible solution for I .The cost of inconsistent planning is defined as the ratio (cid:37) = c ( S ) opt where S = { x , x , . . . , x k } is the solution returned by the agent, and opt = c ( S opt ) is the cost of an optimal solution S opt for the same instance of the considered minimization problem. Approximated evaluation.
It can happen that the considered minimization problem iscomputationally hard, say NP-hard, and the agent is unable to compute a feasible solution S of minimum cost c β ( S ) exactly. Then the agent can pick an approximate solution instead. Forthis situation, we modify the above strategy of the agent as follows. Assume that the agent hasaccess to an α -approximation algorithm A that, given an instance I , computes a feasible solution S ∗ to the instance such that c β ( S ∗ ) ≤ α min c β ( S ), where minimum is taken over all feasiblesolution S to I . For simplicity, we assume throughout the paper that α ≥ α is a function of the input size or opt .Again, the agent uses an inductive scheme to construct a solution. Initially, I = I . Forevery k ≥
0, given the instance I k , the agent computes a feasible solution S k of cost at most α min c β ( S ), where the minimum is taken over all feasible solutions S of I k . Then, exactly asbefore, the agent finds x k = arg min y ∈ S k ω ( y ). If { x , x , . . . , x k } is a feasible solution for I , thenthe agent stops. Otherwise, we set I k +1 = I k (cid:114) { x k } and proceed. The α -approximative cost ofinconsistent planning is defined as the ratio (cid:37) α = c ( S ) opt where S = { x , x , . . . , x k } . Clearly, the1-approximative cost coincides with (cid:37) . We start with a simple observation. Given a feasible solution S for an instance I of a mini-mization problem, we say that x ∈ S is superfluous in S if S (cid:114) { x } is also feasible for I . Theability for the agent to make superfluous choices yields trivial scenarios in which the cost ra-tio (cid:37) can be arbitrarily large. This is for instance the case of an instance of set-cover, defined7s one set y = { , . . . , n } of weight c > n sets x i = { i } , each ofweight 1, for i = 1 , . . . , n . Every solution S i = { x i , y } is feasible, for i = 1 , . . . , n , and satisfies c β ( S i ) = 1 + βc . As a result, whenever 1 + βc < c , the present-biased agent constructs the so-lution S = { x , . . . , x n } , which yields a cost ratio (cid:37) = n/c , which can be made arbitrarily largeas n grows. Instead, if the agent is bounded to avoid superfluous choices, that is, to systemat-ically choose minimal feasible solutions, then only the feasible solutions { y } and { x , . . . , x n } can be considered. As a result, the agent will compute the optimal solution S opt = { y } if c < β ( n − minimal feasible solutions, i.e.,solutions with no superfluous elements, is not sufficient to avoid procrastination. That is, itdoes not prevent the agent from computing solution with high cost ratio. This is for instancethe case of another instance of set-cover, that we denote by I ( n ) SC for further references. Set-cover instance I ( n ) SC : specified by 2 n subsets of { , . . . , n } defined as x i = { i } withweights 1, and y i = { i, . . . , n } with weight c >
1, for i = 1 , . . . , n .The minimal feasible solutions of I ( n ) SC are { y } of weight c , { x , . . . , x i , y i +1 } of weight i + c for i = 1 , . . . , n −
1, and { x , . . . , x n } of weight n . Whenever 1 + βc < c , a time-biased agentbounded to make non-superfluous choices only yet constructs the solution { x , . . . , x n } whichyields a cost ratio (cid:37) = n/c , which can be made arbitrarily large as n grows. We need thefollowing lemma about biased solutions for minimization problems. Lemma 1.
Let α ≥ and let S ∗ be a feasible solution for minimization problem, satisfying c β ( S ∗ ) ≤ α min c β ( S ) , where the minimum is taken over all feasible solutions. Then(i) ω ( x ) ≤ α · opt for x = arg min y ∈ S ∗ ω ( y ) , and(ii) c ( S ∗ ) ≤ αβ opt .Proof. Let S be an optimum solution. As β <
1, it follows that ω ( x ) ≤ ω ( x ) + β · ω ( S ∗ \{ x } ) = c β ( S ∗ ) ≤ α · c β ( S ) ≤ α · c ( S ) = α · opt , and this proves (i). To show (ii), note that c ( S ∗ ) = ω ( x ) + ω ( S ∗ \ { x } ) = β ( βω ( x ) + βω ( S ∗ \ { x } )) , from which it follows that c ( S ∗ ) ≤ β ( ω ( x ) + βω ( S ∗ \ { x } )) = β c β ( S ∗ ) ≤ αβ c β ( S ) ≤ αβ c ( S ) = αβ opt , which completes the proof.Lemma 1 has a simple consequence that also can be derived from the results of Gravin etal. [8, Claim 5.1], that we state as a theorem despite its simplicity, as it illustrates one majordifference between our model and the model in [11]. Theorem 1.
For every α ≥ and every minimization problem, the α -approximative cost ra-tio (cid:37) α cannot exceed α · k where k is the number of steps performed by the agents to constructthe feasible solution { x , . . . , x k } by following the time-biased strategy.Proof. By Lemma 1(i), at any step i ≥ x i ∈ S I in the current partial solution, and this element satisfies ω ( x i ) ≤ α c β ( S opt ) ≤ α c ( S opt ) = α · opt . Therefore, if the agent computes a solution { x , . . . , x k } , then the α -approximative costratio for this solution satisfies (cid:37) α = (cid:80) ki =1 ω ( x i ) / opt ≤ α k, as claimed. Remark.
The bound in Theorem 1 is in contrast to the general model in [11], in which anagent performing k steps can incur a cost ratio exponential in k . This is because the modelin [11] enables to construct graphs with arbitrary weights. In particular, in a graph such asthe one depicted on Fig. 1, one can set up weights such that the weight of ( v , t ) is a constanttime larger than the weight of ( s, t ), the weight of ( v , t ) is in turn a constant time larger8han the weight of ( v , t ), etc., and still a present-biased agent starting from s would travel via v , v , . . . , v k before reaching t . In this way, the sum of the weights of the edges traversed bythe agent may become exponential in the number of traversed edges. This phenomenon doesnot occur when focussing on minimization tasks. Indeed, given a partial solution, the cost ofcompleting this solution into a global feasible solution cannot exceed the cost of constructing aglobal feasible solution from scratch.It follows from Theorem 1 that I ( n ) SC is a worst-case instance. Interestingly, this instance fitswith realistic procrastination scenarios in which the agent has to perform a task (e.g., learning ascientific topic T ) by either energetically embracing the task (e.g., by reading a single thick bookon topic T ), or starting first by an easier subtask (e.g., by first reading a digest of a subtopicof topic T ), with the objective of working harder later, but underestimating the cost of thispostponed hard work. The latter strategy may result in procrastination, by performing a verylong sequence of subtasks x , x , . . . , x n .In fact, I ( n ) SC appears to be the essence of procrastination in the framework of minimiza-tion problems. Indeed, we show that if the cost ratio is large, then the considered instance I contains an instance of the form I ( n ) SC with large n . More precisely, we say that an instance I contains an instance J as a minor if the ground set S J associated to J is a collectionof subsets of the ground set S I associated to I , that is S J ⊆ S I , and, for every ¯ S ⊆ S J ,¯ S is feasible for J if and only if S = (cid:83) ¯ x ∈ ¯ S ¯ x is feasible for I . Moreover, the weight function¯ ω for the elements of S J must be induced by the one for S I as ¯ ω (¯ x ) = (cid:80) x ∈ ¯ x ω ( x ) for ev-ery ¯ x ∈ S J . Let J ( n ) be any instance of a minimization problem such that its associatedground set is S J ( n ) = { x , . . . , x n } ∪ { y , . . . , y n } , and the set of feasible solutions for J ( n ) is F ( J ( n ) ) = (cid:8) { y } , { x , y } , { x , x , y } , . . . , { x , . . . , x n − , y n } , { x , . . . , x n } (cid:9) . The following result sheds some light on why the procrastination structure ofFig. 1 pops up.
Theorem 2.
Let I be an instance of a minimization problem for which the present-biased agentwith parameter β ∈ (0 , computes a solution for I with cost α · opt ( I ) for some α > . Then I contains J ( n ) as a minor for some n ≥ α , and the present-biased agent with parameter β computes a solution for J ( n ) with cost α · opt ( J ( n ) ) .Proof. Let S = { x , . . . , x n } be the final solution selected by the present-biased agent for I , andlet ω be the weight function on the set S I associated to I . We have (cid:80) ni =1 ω ( x i ) = α opt ( I ).For every i ∈ { , . . . , n } , let us denote by opt ( I (cid:114) { x , . . . , x i } ) the cost of an optimal solutionfor the instance I (cid:114) { x , . . . , x i } , and by S opt ( I (cid:114) { x ,...,x i } ) a corresponding optimal solution. For i = 0, S opt ( I (cid:114) { x ,...,x i } ) is an optimal solution for I . For i = 1 , . . . , n , we define¯ x i = { x i } , and ¯ y i = S opt ( I (cid:114) { x ,...,x i − } ) . Let J be the instance with ground set { ¯ x , . . . , ¯ x n } ∪ { ¯ y , . . . , ¯ y n } , and feasible solutions { ¯ y } , { ¯ x , ¯ y } , . . . , { ¯ x , . . . , ¯ x n − , ¯ y n } , { ¯ x , . . . , ¯ x n } . Note that ¯ x i (cid:54) = ¯ x j for every i (cid:54) = j , because x i (cid:54) = x j for every i (cid:54) = j . Also, for every i ∈{ , . . . , n − } and j ∈ { , . . . , n } , ¯ x i (cid:54) = ¯ y j , because otherwise the sequence constructed by thetime-biased agent for I would stop at x k with k < n . Therefore, we have J = J ( n ) , and, since ω ( x i ) ≤ opt ( I ) for every i = 1 , . . . , n , n ≥ α holds.For analyzing the behavior of a present-biased agent with parameter β acting on J , letus assume that k steps were already performed by the agent, with 0 ≤ k < n , resulting inconstructing the partial solution { ¯ x , . . . , ¯ x k } . (For k = 0, this partial solution is empty). Thefeasible solutions for J k = J (cid:114) { ¯ x , . . . , ¯ x k } are { ¯ y } , . . . , { ¯ y k +1 } , { ¯ x k +1 , ¯ y k +2 } , . . . , { ¯ x k +1 , . . . , ¯ x n − , ¯ y n } , { ¯ x k +1 , . . . , ¯ x n } . i ∈ { , . . . , n } , ¯ ω (¯ x i ) = ω ( x i ), and ¯ ω (¯ y i ) = opt ( I (cid:114) { x , . . . , x i − } ). Weclaim that ¯ x k +1 is the next element chosen by the agent. Indeed, note first that ¯ ω (¯ x k +1 ) ≤ ¯ ω (¯ y k +2 ), as, otherwise, we would get ω ( x k +1 ) > opt ( I (cid:114) { x , . . . , x k +1 } ), contradicting thechoice of x k +1 by the agent performing on I . As a consequence, c β ( { ¯ x k +1 , ¯ y k +2 } ) = ¯ ω (¯ x k +1 ) + β ¯ ω (¯ y k +2 ) . It follows from the above that, for every j = 1 , . . . , k + 1, c β ( { ¯ y j } ) ≥ c β ( { ¯ x k +1 , ¯ y k +2 } ), as thereverse inequality would contradict the choice of x k +1 by the agent performing on I . For thesame reason, for every (cid:96) ∈ { k + 1 , . . . , n − } , and every i ∈ { k + 1 , . . . , (cid:96) } , we have c β ( { ¯ x k +1 , ¯ y k +2 } ) ≤ ¯ ω (¯ x i ) + β (cid:16) (cid:88) j ∈{ k +1 ,...,(cid:96) } (cid:114) { i } ¯ ω (¯ x j ) + ¯ ω (¯ y (cid:96) +1 ) (cid:17) and c β ( { ¯ x k +1 , ¯ y k +2 } ) ≤ ¯ ω (¯ y (cid:96) +1 ) + β (cid:88) j ∈{ k +1 ,...,(cid:96) } ¯ ω (¯ x j ) . As a consequence, the present-biased agent performing on J picks ¯ x k +1 at step k + 1, as claimed.The cost of the solution computed by the agent is (cid:80) ni =1 ¯ ω (¯ x i ) = (cid:80) ni =1 w ( x i ) = α opt ( I ). Onthe other hand, by construction, opt ( J ) = ¯ ω (¯ y ) = opt ( I ). The cost ratio of the solutioncomputed by the agent for J is thus α , which completes the proof. In the previous section, we have observed that forcing the agent to avoid superfluous choices,by picking minimal feasible solutions only, does not prevent it from constructing solutions thatare arbitrarily far from the optimal. In this section, we show that, by enforcing consistency inthe sequence of partial solutions constructed by the agent, such bad behavior does not occur.More specifically, given a feasible solution S for I , we say that x is inconsistent with S if x / ∈ S .The following result shows that inconsistent choices is what causes high cost ratio. Theorem 3.
An agent using an α -approximation algorithm bounded to avoid inconsistentchoices with respect to the feasible solutions used in the past for constructing the current partialsolution returns an α/β -approximation of the optimal solution. This holds independently fromwhether the agent makes superfluous choices or not.Proof. Let I be an instance of a minimization problem ( I , F, c ). Let S = { x , . . . , x k } be thesolution constructed by the agent for I , where x i is the element computed by the agent at step i ,for i = 0 , . . . , k . Let S i be the feasible solution of I i = I (cid:114) { x , . . . , x i − } considered by theagent at step i . Since the agent is bounded to avoid any inconsistent choices with respect tothe past, we have x i ∈ ∩ ij =0 S j for every i = 0 , . . . , k because x i / ∈ S j for some j < i would bean inconsistent choice. It follows that S ⊆ S . Therefore, c ( S ) ≤ c ( S ). Since the agent uses an α -approximation algorithm, by Lemma 1(ii), c ( S ) ≤ αβ opt and the claim follows. We first investigate the cost ratio for minimization problems for the case when β >
1. Similarbound was obtained by Kleinberg et al. (see [12, Theorem 2.1]). However, their theorem isabout sophisticated agents and cannot be applied in our case directly.
Theorem 4.
Solutions computed by present-biased agents satisfy the following: For any mini-mization problem with β > , the cost ratio is at most β . roof. For the proof of the theorem it is convenient to switch to the original graph-theoreticmodel of Kleinberg and Oren [11]. Note that the task graphs corresponding to optimizationproblems are, in fact, directed acyclic graphs. Hence, we only consider task graphs of this typeto avoid dealing with paths of maximum length in the presence of cycles.Let G be a directed acyclic graph (DAG) with a source s . Let also ω : E ( G ) → N be a weightfunction. The aim of the agent is to go from the source s to a sink t of G making present-biaseddecisions on each step. We assume that G has an s - t path. Let β be a positive constant distinctfrom 1. Let c min ( x ) be the minimum length of an x - t path.We suppose that the agent is equipped with an algorithm A that, given a vertex v ∈ V ( G ),finds a vertex x ∗ ∈ N + G ( v ) such that ω ( vx ∗ ) + β · c min ( x ∗ ) = min x ∈ N + G ( v ) ( ω ( vx ) + β · c min ( x )) . The agent constructs an s - t path as follows: if the agent occupies a vertex v (cid:54) = t , then he makesthe present-biased α -approximate estimation of the length of a shortest v - t path and moves to x ∗ . Note that since G is a DAG, the agent would eventually arrive to t .We denote by cost min ( v ) the length of a v - t path constructed by the agent from v . Noticethat this value is not uniquely defined as the agent may be able to choose distinct verticesthat provide α -approximate present-biased evaluations but could give distinct lengths for theconstructed paths. Then the proof of the theorem is implied by the following claim. Claim 3.1.
Let G be a weighted DAG with a weigh function ω : E ( G ) → N and a sink t . Thenfor every v ∈ V ( G ) , if β > , then cost min ( v ) ≤ β · c min ( v ) . The claim is trivial if v = t . Assume that v (cid:54) = t , and that the claim holds for everyout-neighbor x of v . Assume that x ∗ ∈ N + G ( v ) is computed by A , and let y = arg min x ∈ N + G ( v ) ( ω ( vx ) + c min ( x )) . That is, there is a shortest v - t path that goes through y .By induction, we have that cost min ( x ∗ ) ≤ β · c min ( x ∗ ) . It follows that cost min ( v ) = ω ( vx ∗ ) + cost min ( x ∗ ) ≤ ω ( vx ∗ ) + β · c min ( x ∗ ) ≤ ω ( vx ∗ ) + β · c min ( x ∗ ) = min x ∈ N + G ( v ) ( ω ( vx ) + β · c min ( x )) ≤ ω ( vy ) + β · c min ( y ) ≤ β · ω ( vy ) + β · c min ( y ) ≤ β · c min ( v ) . The last inequality completes the proof of the claim, and of the theorem.Next, we consider maximization problems. The formalism for these variants can be set upin a straightforward manner by adapting the framework displayed in Section 2.1. We establishthe following worst-case bounds.
Theorem 5.
Solutions computed by present-biased agents satisfy the following:(i) For any maximization problem with β < , the cost ratio is at most β ;(ii) For any maximization problem with β > , the cost ratio is at most β c , where c ≤ opt isthe cost of a solution constructed by the agent. roof. As in the proof of Theorem 4, we switch to the graph-theoretic model of Kleinbergand Oren [11]. Let G be a directed acyclic graph (DAG) with a source s and weight function ω : E ( G ) → N . The agent aims to go from the source s to a sink t of G making present-biaseddecisions on each step. We assume that G has an s - t path. Let β be a positive constant distinctfrom 1, and let c max ( x ) be the maximum length of an x - t path.Let α ∈ (0 , A that, given a vertex v ∈ V ( G ), finds a vertex x ∗ ∈ N + G ( v ) such that ω ( vx ∗ ) + β · c max ( x ∗ ) = max x ∈ N + G ( v ) ( ω ( vx ) + β c max ( x )) . Using A , the agent located in vertex v (cid:54) = t constructs an s - t path as follows: the agent computesthe present-biased α -approximate estimation of the length of a longest v - t and moves to x ∗ . Wedenote by cost max ( v ) the length of a v - t path constructed by the agent from v . Claim 3.2.
Let G be a weighted DAG with a weigh function ω : E ( G ) → N and a sink t . Thenfor every v ∈ V ( G ) , (i) if β < , then cost max ( v ) ≥ β c max ( v ) , (ii) if β > , then c max ( v ) ≤ cost max ( v ) β cost max ( v ) . As in Theorem 4, we prove the claim by induction. The claim is trivial if v = t . Assume that v (cid:54) = t , and that the claim holds for every out-neighbor x of v . Let x ∗ be the vertex computedby A and let y = arg max x ∈ N + G ( v ) ( ω ( vx ) + c max ( x )) . That is, there is a longest v - t path that goes through y .To show (i), we use the inductive assumption that cost max ( x ∗ ) ≥ β c max ( x ∗ ) . We have cost max ( v ) = ω ( vx ∗ ) + cost max ( x ∗ ) ≥ ω ( vx ∗ ) + β c max ( x ∗ )= max x ∈ N + G ( v ) ( ω ( vx ) + β c max ( x )) ≥ ω ( vy ) + β c max ( y ) ≥ β ω ( vy ) + β c max ( y ) ≥ β c max ( v ) . To prove (ii), we assume that the following inductive assumption holds: c max ( x ∗ ) ≤ cost max ( x ∗ ) β cost max ( x ∗ ) . It follows that c max ( v ) = ω ( vy ) + c max ( y ) ≤ ω ( vy ) + β · c max ( y ) ≤ max x ∈ N G ( v ) ( ω ( vx ) + β · c max ( x )) ≤ ω ( vx ∗ ) + β · c max ( x ∗ ) ≤ ω ( vx ∗ ) + β · cost max ( x ∗ ) β cost max ( x ∗ ) ≤ β cost max ( x ∗ )+1 ( ω ( vx ∗ ) + cost max ( x ∗ )) = cost max ( v ) β cost max ( x ∗ )+1 ≤ cost max ( v ) β cost max ( v ) . This last inequality completes the proof of Claim 3.2, which immediately gives the bounds forthe α -approximate cost ratio claimed in the statement of the theorem.12e also can write the bound for the cost ratio for β > opt . Corollary 1.
For any maximization problem with β > , the cost ratio is at most (1 +log β ) opt log opt .Proof. Let c be the cost of a solution constructed by the agent. By Theorem 5, opt ≤ cβ c .Therefore, log opt ≤ log c + c log β ≤ (cid:0) β ) c, and opt c ≤ (1 + log β ) opt log opt . For minimization problems with β >
1, and maximization problems with β <
1, we have thatthe cost ratio is bounded by a constant. This differs drastically with the case of maximizationproblems with β >
1, when the cost ratio is still bounded but the bound is exponential. Thisexponential upper bound is however essentially tight, in the sense that the exponent cannot beavoided.
Theorem 6.
There are maximization problems for which a present-biased agent with β > returns a solution whose cost ratio is at least c β c − , where c is the cost of the solutionconstructed by the agent.Proof. Let us consider the maximum independent set problem. In this problem, we are given aweighted graph G , and the task is to find an independent set of maximum weight. Let k be apositive integer. We construct the graph G k as follows (see Fig. 2): • construct k + 1 vertices x , . . . , x k , and make them pairwise adjacent, • construct k vertices y , . . . , y k , • for each i ∈ { , . . . , k } , make y i adjacent to x i , x i +1 , . . . , x k .To define the weights, let β ≥
2. We set ω ( y i ) = 1 for every i ∈ { , . . . , k } , and ω ( x i ) = β i forevery i ∈ { , . . . , k } . clique y y y y x x x x x β β β β β Figure 2: Construction of G k for k = 4.Since X = { x , . . . , x k } is a clique, any independent set has at most one vertex in X .Therefore, the family of sets S i = { x i } ∪ { y i +1 , . . . , y k } for i ∈ { , . . . , k } form the family of maximal independent sets. Because β ≥
2, it is straightfor-ward to verify that the single-vertex set S k = { x k } is an independent set of maximum weight β k , that is, opt = β k . Observe that the biased cost of this set is β k as well.From the other side, the biased cost of S k − is ω ( y k ) + β · ω ( x k − ) = 1 + β · β k − = 1 + β k > β k . Hence, the agent would prefer to select y k at the first iteration. At the next iteration, the agentconsiders the graph obtained from G k by the deletion of y k and its neighborhood, that is, G k − .Applying the same arguments inductively, we conclude that the agent will end up with the set S = { x , y , . . . , y k } with ω ( S ) = k + 1. We obtain that opt = β c − , where c is the cost of asolution constructed by the agent. 13 emark. An example similar to the one in the proof of Theorem 6 can be constructed forthe knapsack problem. Recall that in this problem, we are given n objects with positive integer values v i , and weights w i , for i ∈ { , . . . , n } , and W ∈ N . the task is to find a set of objects S ⊆ { , . . . , n } of maximum value with the total weight at most W . Let k be a positive integer.We let n = 2 k + 1, W = n and β ≥
2. We define v i = β k +1 − i , and w i = W − ( i − i ∈ { , . . . , k + 1 } , and we set v i = 1 and w i = 1 for i ∈ { k + 2 , . . . , n } . Using the samearguments as for the maximum independent set problem, we obtain that the optimum solutionhas cost β k while a present-biased agent would select a solution of cost k + 1. In Section 2.2, we have seen several instances of the set-cover problem whose cost ratio cannotbe bounded by any function of opt . The same obviously holds for the hitting-set problem.Recall that an instance of hitting-set is defined by a collection Σ of subsets of a finite set V , andthe objective is to find the subset S ⊆ V of minimum size, or minimum weight, which intersects(hits) every set in Σ. However, set-cover problems, and hitting set problems behave differentlywhen the sizes of the sets are bounded. First, we consider the d -set cover problem. The d -set cover problem. Let d be a positive integer. The task of the d -set cover problemis, given a collection Σ of subsets with size at most d of a finite set V , and given a weightfunction ω : Σ → N , find a set S ⊆ Σ of minimum weight that covers V , that is, (cid:83) X ∈ S X = V . Theorem 7.
Let α ≥ . For any instance of the d -set-cover problem, the α -approximative costratio is at most α · d · opt .Proof. Let I = (Σ , V, ω ) be an instance of the d -set cover problem. Let | V | = n . Denote by S , . . . , S p a sequence of solutions computed by the present-biased agent avoiding superfluoussolutions, and let S = { X , . . . , X p } be the obtained solution for I . That is, S is a set coversuch that X i = arg min Y ∈ S i ω ( Y )for i ∈ { , . . . , p } . Clearly, p ≤ n . Note that for each iteration i ∈ { , . . . , p } , the agent considersthe instance I i = (Σ i , V i , ω i ), where, as superfluous solutions are avoided, V i = V \ ( (cid:83) i − j =1 X i ) , Σ i = { X ∩ V i | X ∈ Σ \ { X , . . . , X i − }} ,ω i = ω | Σ i . We have that, for every i ∈ { , . . . , p } , opt = opt ( I ) = opt ( I ) ≥ · · · ≥ opt ( I p ) , and, by Lemma 1(ii), ω ( X i ) ≤ α opt ( I i ) . Since each set of Σ covers at most d elements of V , opt ( I ) ≥ n/d . Therefore c ( S ) = p (cid:88) i =1 ω ( X i ) ≤ α opt ( I ) n ≤ αd opt ( I ) . It follows that the cost ratio is at most αd opt .14 he d -hitting set problem. Let d be a positive integer. We are given a collection Σ ofsubsets with size d of a finite set V , a weight function ω : V → N . The task is to find a set S ⊆ V of minimum weight that hits every set of Σ.We are using the classical sunflower lemma of Erd˝os and Rado [6]. We state this result inthe form given in [5]. A sunflower with k petals and a core X is a collection of pairwise distinctsets S , . . . , S k such that S i ∩ S j = X for all distinct i, j ∈ { , . . . , k } . Note that the core maybe empty, that is, a collection of k pairwise disjoint sets is a sunflower. Lemma 2 (Sunflower lemma [6]) . Let A be a family of pairwise distinct sets over a universe U such that for every A ∈ A , | A | = d . If |A| > d !( k − d , then A contains a sunflower with k petals. Theorem 8.
Let α ≥ . For any instance of the d -hitting-set problem, the α -approximativecost ratio is at most αd ! ( αβ opt ) d .Proof. Let I = (Σ , V, ω ) be an instance of the d -hitting set problem. Denote by S , . . . , S p asequence of solutions computed by the present-biased agent avoiding superfluous solutions, andlet S = { v , . . . , v p } be the obtained solution for I , that is, S ⊆ V is a hitting set such that v i = arg min v ∈ S i ω ( v )for every i ∈ { , . . . , p } . Since the agent avoids superfluous solutions, the agent considers theinstance I i = (Σ i , V i , ω i ) at each iteration i ∈ { , . . . , p } , where V i = V \ { v , . . . , v i − } , Σ i = { X ∈ Σ | X ∩ { v , . . . , v i − } = ∅} ,ω i = ω | V i . Let i ∈ { , . . . , p } . Since S i is a minimal hitting set for Σ i , there is X i ∈ Σ i such that v i ∈ X i ,and, for every v ∈ S i \ { v i } , v / ∈ X i . We say that X i is a private set for v i . Observe that opt = opt ( I ) = opt ( I ) ≥ · · · ≥ opt ( I p )by the construction of the instances. The following claim is crucial for the prof of the theorem. Claim 4.1. p ≤ d ! ( αβ opt ) d . Let us assume, for the purpose of contradiction, that p > d ! ( αβ opt ) d . Consider the privatesets X , . . . , X p ∈ Σ for v , . . . , v p , respectively, and let A = { X , . . . , X p } . Note that X , . . . , X p are pairwise distinct, since X h ∈ Σ h and X h / ∈ Σ h +1 for every h ∈ { , . . . , p − } . Let k = (cid:106) αβ opt (cid:107) + 1 ≥ . We have that | X h | = d for every X h ∈ A , and |A| > d ! ( k − d . Hence, A contains a sunflowerwith k petals by Lemma 2. Denote by X h , . . . , X h k the sets of the sunflower, and let Y be itscore.Suppose that Y = ∅ . Then, X h , . . . , X h k are pairwise disjoint. Therefore, opt ( I ) ≥ k = (cid:106) αβ opt (cid:107) + 1 > opt , which is a contradiction. Therefore, Y (cid:54) = ∅ .We show that, for every hitting set R for Σ h of weight at most αβ opt , R ∩ Y (cid:54) = ∅ . Assumethat this is not the case, that is, R ∩ Y = ∅ . Since R is a hitting set, there exists u (cid:96) ∈ X h (cid:96) \ Y u (cid:96) ∈ R for every (cid:96) ∈ { , . . . , k } . Because { X h , . . . , X h k } is a sunflower, u , . . . , u k aredistinct. It follows that ω ( R ) ≥ k (cid:88) (cid:96) =1 ω ( u (cid:96) ) ≥ k = (cid:106) αβ opt (cid:107) + 1 > αβ opt . The latter strict inequality is contradicting the fact that the weight of R is at most αβ opt . Weconclude that R ∩ Y (cid:54) = ∅ .Recall that S h is a feasible solution for I h , and, by Lemma 1 (ii), ω ( S h ) ≤ αβ opt ( I h ) ≤ αβ opt . Then S h ∩ Y (cid:54) = ∅ . The set X h was chosen to be a private set for v h , that is, S h ∩ X h = { v h } .Note that v h / ∈ X h . Hence, v h / ∈ Y , and thus S h ∩ Y = ∅ . This is a contradiction, whichcompletes the proof of the claim.By Claim 4.1, p ≤ d ! ( αβ opt ) d . By Lemma 1 (i), ω ( v i ) ≤ α opt ( I i ) ≤ α opt for every i ∈ { , . . . , p } . Therefore, c ( S ) = p (cid:88) i =1 ω ( v i ) ≤ α opt p ≤ α opt d ! ( αβ opt ) d . It follows that c ( S ) opt ≤ αd ! ( αβ opt ) d .Let S = { x , . . . , x n } be the final solution selected by the present-biased agent for I , andlet ω be the weight function on the set S I associated to I . We have (cid:80) ni =1 ω ( x i ) = α opt ( I ).For every i ∈ { , . . . , n } , let us denote by opt ( I (cid:114) { x , . . . , x i } ) the cost of an optimal solutionfor the instance I (cid:114) { x , . . . , x i } , and by S opt ( I (cid:114) { x ,...,x i } ) a corresponding optimal solution. For i = 0, S opt ( I (cid:114) { x ,...,x i } ) is an optimal solution for I . For i = 1 , . . . , n , we define¯ x i = { x i } , and ¯ y i = S opt ( I (cid:114) { x ,...,x i − } ) . Let J be the instance with ground set { ¯ x , . . . , ¯ x n } ∪ { ¯ y , . . . , ¯ y n } , and feasible solutions { ¯ y } , { ¯ x , ¯ y } , . . . , { ¯ x , . . . , ¯ x n − , ¯ y n } , { ¯ x , . . . , ¯ x n } . Note that ¯ x i (cid:54) = ¯ x j for every i (cid:54) = j , because x i (cid:54) = x j for every i (cid:54) = j . Also, for every i ∈{ , . . . , n − } and j ∈ { , . . . , n } , ¯ x i (cid:54) = ¯ y j , because otherwise the sequence constructed by thetime-biased agent for I would stop at x k with k < n . Therefore, we have J = J ( n ) , and, since ω ( x i ) ≤ opt ( I ) for every i = 1 , . . . , n , n ≥ α holds.For analyzing the behavior of a present-biased agent with parameter β acting on J , letus assume that k steps were already performed by the agent, with 0 ≤ k < n , resulting inconstructing the partial solution { ¯ x , . . . , ¯ x k } . (For k = 0, this partial solution is empty). Thefeasible solutions for J k = J (cid:114) { ¯ x , . . . , ¯ x k } are { ¯ y } , . . . , { ¯ y k +1 } , { ¯ x k +1 , ¯ y k +2 } , . . . , { ¯ x k +1 , . . . , ¯ x n − , ¯ y n } , { ¯ x k +1 , . . . , ¯ x n } . Note that, for every i ∈ { , . . . , n } , ¯ ω (¯ x i ) = ω ( x i ), and ¯ ω (¯ y i ) = opt ( I (cid:114) { x , . . . , x i − } ). Weclaim that ¯ x k +1 is the next element chosen by the agent. Indeed, note first that ¯ ω (¯ x k +1 ) ≤ ¯ ω (¯ y k +2 ), as, otherwise, we would get ω ( x k +1 ) > opt ( I (cid:114) { x , . . . , x k +1 } ), contradicting thechoice of x k +1 by the agent performing on I . As a consequence, c β ( { ¯ x k +1 , ¯ y k +2 } ) = ¯ ω (¯ x k +1 ) + β ¯ ω (¯ y k +2 ) .
16t follows from the above that, for every j = 1 , . . . , k + 1, c β ( { ¯ y j } ) ≥ c β ( { ¯ x k +1 , ¯ y k +2 } ), as thereverse inequality would contradict the choice of x k +1 by the agent performing on I . For thesame reason, for every (cid:96) ∈ { k + 1 , . . . , n − } , and every i ∈ { k + 1 , . . . , (cid:96) } , we have c β ( { ¯ x k +1 , ¯ y k +2 } ) ≤ ¯ ω (¯ x i ) + β (cid:16) (cid:88) j ∈{ k +1 ,...,(cid:96) } (cid:114) { i } ¯ ω (¯ x j ) + ¯ ω (¯ y (cid:96) +1 ) (cid:17) and c β ( { ¯ x k +1 , ¯ y k +2 } ) ≤ ¯ ω (¯ y (cid:96) +1 ) + β (cid:88) j ∈{ k +1 ,...,(cid:96) } ¯ ω (¯ x j ) . As a consequence, the present-biased agent performing on J picks ¯ x k +1 at step k + 1, as claimed.The cost of the solution computed by the agent is (cid:80) ni =1 ¯ ω (¯ x i ) = (cid:80) ni =1 w ( x i ) = α opt ( I ). Onthe other hand, by construction, opt ( J ) = ¯ ω (¯ y ) = opt ( I ). The cost ratio of the solutioncomputed by the agent for J is thus α , which completes the proof. We demonstrated that, by focussing on present-biased agents solving tasks, specific detailedanalysis can be carried on for each considered task, which enables to identify very differentagent’s behavior depending on the tasks (e.g., set cover vs. hitting set). Second, focussingon present-biased agents solving tasks enables to generalize the study to overestimation, andto maximization, providing a global picture of searching via present-biased agents. Yet, lotsremain to be done for understanding the details of this picture.In particular, efforts could be made for studying other specific classical problems in thecontext of searching by a present-biased agent. This includes classical optimization problemslike traveling salesman (TSP), metric TSP, maximum matching, feedback vertex set, etc. Suchstudy may lead to a better understanding of the class of problems for which present-biasedagents are efficient, and the class for which they act poorly. And, for problems for whichpresent-biased agents are acting poorly, it may be of high interest to understand what kind ofrestrictions on the agent’s strategy may help the agent finding better solutions.Another direction of research is further investigation of the influence of using approximationalgorithms by agents, as it is natural to assume that the agents are unable compute the costexactly. We made some initial steps in this direction, but it seems that this area is almostunexplored. For example, it can be noted that the upper bound for the cost ratio in Theorem 4can be rewritten under the assumption that the agent uses an α -approximation algorithm.However, the bound gets blown-up by the factor α s , where s is the size of the solution obtainedby the agent (informally, we pay the factor α on each iteration). From the other side, theexamples in Section 4 show that this is not always so. Are there cases when this exponentialblow-up unavoidable? The same question can be asked about maximization problems. References [1]
G. A. Akerlof , Procrastination and obedience , American Economic Review: Papers andProceedings, 81 (1991), pp. 1–19. 1, 2, 3, 5[2]
S. Albers and D. Kraft , On the value of penalties in time-inconsistent planning , in 44thInternational Colloquium on Automata, Languages, and Programming (ICALP), 2017,pp. 10:1–10:12. 6[3] ,
The price of uncertainty in present-biased planning , in 13th Int. Conference on Weband Internet Economics (WINE), 2017, pp. 325–339. 6174] ,
Motivating time-inconsistent agents: A computational approach , Theory Comput.Syst., 63 (2019), pp. 466–487. 6[5]
M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk,M. Pilipczuk, and S. Saurabh , Parameterized Algorithms , Springer, 2015. 15[6]
P. Erd˝os and R. Rado , Intersection theorems for systems of sets , J. London Math. Soc.,35 (1960), pp. 85–90. 5, 15[7]
N. Gravin, N. Immorlica, B. Lucier, and E. Pountourakis , Procrastination withvariable present bias , in ACM Conference on Economics and Computation (EC), 2016,p. 361. 6[8] ,
Procrastination with variable present bias , CoRR, abs/1606.03062 (2016). 6, 8[9]
C. Jepson, G. Loewenstein, and P. Ubel , Actual versus estimated differences in qual-ity of life before and after renal transplant . Working Paper, Department of Social andDecision Sciences, Carnegie Mellon University, 2001. 3[10]
J. M. Kleinberg and S. Oren , Time-inconsistent planning: a computational problemin behavioral economics , in ACM Conference on Economics and Computation (EC), 2014,pp. 547–564. 1, 5, 6[11] ,
Time-inconsistent planning: a computational problem in behavioral economics , Com-mun. ACM, 61 (2018), pp. 99–107. 1, 2, 3, 4, 5, 8, 11, 12[12]
J. M. Kleinberg, S. Oren, and M. Raghavan , Planning problems for sophisticatedagents with present bias , in ACM Conference on Economics and Computation (EC), 2016,pp. 343–360. 6, 10[13] ,
Planning with multiple biases , in ACM Conference on Economics and Computation(EC), 2017, pp. 567–584. 6[14]
G. Loewenstein, T. O’Donoghue, and M. Rabin , Projection bias in predicting futureutility , The Quarterly Journal of economics, 118 (2003), pp. 1209–1248. 3[15]