Penalty Bidding Mechanisms for Allocating Resources and Overcoming Present BiasThe authors would like to thank Ido Erev, Matt Juszczak, Scott Kominers, and Kyle Pasake for helpful comments and discussions.

Penalty Bidding Mechanisms for Allocating Resources and Overcoming Present Biasthanks: The authors would like to thank Ido Erev, Matt Juszczak, Scott Kominers, and Kyle Pasake for helpful comments and discussions.

Hongyao Ma John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA. Email: hongyaoma@seas.harvard.edu.    Reshef Meir Department of Industrial Engineering and Management, Technion - Israel Institute of Technology, Technion City, Haifa 3200003, Israel. Email: reshefm@ie.technion.ac.il.    David C. Parkes John A. Paulson School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Maxwell Dworkin 229, Cambridge, MA 02138, USA. Email: parkes@eecs.harvard.edu.    Elena Wu-Yan Harvard College, Cambridge, MA, 02138, USA. Email: elenaw@college.harvard.edu.
Abstract

From skipped exercise classes to last-minute cancellation of dentist appointments, underutilization of reserved resources abounds, arising from uncertainty about the future and further exacerbated by present bias— the constant struggle between our current and future selves. The contingent second price mechanism proposed by Ma et al. [23] provably optimizes utilization, but relies on the assumption that agents rationally optimize their expected utility, and fails to be incentive compatible when agents are present-biased. In this paper, we unite resource allocation and commitment devices through the design of contingent payment mechanisms. We propose the two-bid penalty-bidding mechanism, which assigns the resources to agents who are willing to accept the highest no-show penalties, determines a market-clearing penalty level, but allows each assigned agent to increase her own penalty in order to better counter her present bias. We prove the existence of a simple dominant strategy equilibrium, regardless of an agent’s level of present bias or degree of “sophistication” (the ability to forecast the change in one’s future preferences). Via simulations, we show that the proposed mechanism not only improves utilization, but also achieves higher social welfare than mechanisms that are welfare-optimal for settings without present bias.

1 Introduction

“It was a disaster,” recalled Matt Juszczak, co-founder of Turnstyle Cycle and Bootcamp, a fitness company that offers cycling and bootcamp classes across five studios in the Boston area. “When we opened our first indoor cycling location in Boston’s Back Bay, we saw 40 to 50 no-shows and late cancels in an average day— that’s over 15,000 in a year!”111https://business.mindbody.io/education/blog/tips-reduce-no-shows-and-late-cancels-your-fitness-business, visited September 1, 2018. Like many well-known exercise franchises, Turnstyle allowed customers to reserve class spots several days in advance with a first-come-first-serve reservation system. However, ambitious customers, overestimating the amount of time in their schedules or their desire to exercise in the future, often snag a spot only to ultimately cancel last-minute or simply not show up.

Figure 1: Log in page of the squash court reservation system at the Harvard Hemenway Gymnasium.

Similarly, the squash courts at Harvard’s Hemenway Gymnasium used to allow students and faculty to reserve time-slots to play squash up to seven days ahead of time. Even though the reservation window has since been reduced to three days because of high no-show rates, the gym operators still feel the need to display the warning shown in Figure 1 every time someone logs into the reservation system.222https://recreation.gocrimson.com/recreation/facilities/Hemenway, visited September 1, 2018. For examples from other domains, organizers of free events report to Eventbrite that their no-show rate can be as high as 50%,333https://www.eventbrite.com/blog/asset/ultimate-way-reduce-no-shows-free-events/, visited 5/6/2019. and even for prepaid events organized through Doorkeeper, the fraction of no-shows can be has high as 20%.444https://www.doorkeeper.jp/event-planning/increasing-participants-decreasing-no-shows?locale=en, visited May 6, 2019. Studies of outpatient clinics report that no-shows can range from 23-34%, with no-shows costing an estimated 14% of daily revenue as well as impacting efficiency.555https://jaoa.org/article.aspx?articleid=2671437, visited May 6, 2019.

Common to all these problems is the presence of uncertainty, self-interest and down-stream utilization decisions on the part of participants, together with the interest of a social planner (a gym manager, an event organizer, a clinic) that a resource be used and not wasted. Beyond revenue and efficiency motivations, utilization can have positive externalities, for example members of a cycling studio derive motivation from fellow bikers in class. Complicating the problem is present bias, the constant struggle between our current and future selves. Consider that at the beginning of the week, someone might prefer a spin class over watching TV on Friday, reserving a spot in a class ahead of time, but by the time Friday comes around prefer to just watch TV instead.

Recognizing the problem of low utilization, many reservation systems are now charging a penalty for not showing up: Turnstyle now charges a $20 penalty for missing a scheduled class,666https://kb.turnstylecycle.com/policies/what-is-the-late-cancel-no-show-policy, visited May 6, 2019. patients who miss appointments at hospitals may need to pay a fee that is not covered by insurance,777https://huhs.harvard.edu/sites/default/files/HDS%20New%20Patient%20Welcome%20Letter-eps-converted-to.pdf, visited May 10th, 2018. and organizers of some conferences collect a deposit that is returned only to students who actually attended talks.888https://risingstarsasia2018.ust.hk/guidelines.php, visited May 10th, 2018. These approaches can be viewed as simple, first-come-first serve schemes, for some choice of no-show penalty.

In recent work, Ma et al. [23] model agents’ future value from using a resource as a random variable, and propose the contingent second price mechanism (CSP). The mechanism takes into account the maximum penalty that an individual participant would be willing to face, which provides a good signal for the individual’s reliability. The CSP mechanism has a simple dominant strategy equilibrium, and provably optimizes utilization among a large family of mechanisms with desired properties. The mechanism, however, relies on the assumption that agents rationally optimize their expected utility, does not accommodate the needs of individuals with different levels of present bias, and can even fail to be incentive compatible for present-biased agents since in this case an agent’s expected utility may no longer be monotone in the no-show penalty.

1.1 Our results

In this paper, we unite through contingent payment mechanisms the allocation of scarce resources under uncertainty and the design of commitment devices for overcoming present bias.

We generalize the model proposed in Ma et al. [23], and decompose an agent’s value from using a resource as the sum of a random value experienced at the time of using the resource (modeling for example the opportunity cost and the present pain of going to the gym), and a fixed value that is not gained until later in the future (consider, for example the future benefit from better health). We incorporate the quasi-hyperbolic discounting model for time-inconsistent preferences [20, 25], such that when an agent is making a decision on whether to use a resource, the future benefit from using a resource is discounted by a present bias factor. Agents may also have different levels of “sophistication”— a naive agent believes she does not discount the future, a sophisticated agent is able to perfectly forecast the change in her future preferences, and a partially naive agent resides somewhere in between.

In period 0, an agent’s private information corresponds to the distribution of her immediate value from using the resource, the future value from using the resource, and what she believes to be her present bias factor. A coordination mechanism elicits information from each agent, assigns each of resources to the agents, and may determine both a base payment that an assigned agent always pays, as well as a penalty for each assigned agent in the event of a no-show. In period 1, each assigned agent learns her immediate value from using the resource, and with knowledge of the penalty and the future value, decides whether or not to use the resource.

The two-bid penalty-bidding mechanism works as follows. In period 0, the mechanism elicits a bid from each agent, representing the highest penalty the agent is willing to accept for the option to use the resource for free. The resources are assigned to the highest bidders, and the mechanism asks each assigned agent to report a penalty that is weakly higher than the bid, representing the actual amount she would like the mechanism to charge in the case of a no-show. The main result (Theorem 3) establishes a simple, dominant strategy equilibrium of the two-bid penalty bidding mechanism, regardless of an agent’s value distribution, level of present bias, or degree of sophistication. The mechanism also satisfies voluntary participation and runs without a budget deficit.

We show via simulations that the mechanism not only improves utilization, but also achieves higher social welfare than the standard price auction, which is the welfare-optimal mechanism for settings without present bias. In particular, in a population where agents have different levels of present bias, the more biased agent benefit more under two-bid penalty bidding, in comparison to the outcome under the price auction. While naive agents do not see the value of commitment and generally do not take any commitment device when offered [6, 4], the two-bid penalty bidding mechanism is able to help, since a commitment device is designed through the mechanism, and not accepting a commitment device is not an option.

1.2 Related Work

To the best of our knowledge, this current paper is the first to study resource assignment in the presence of uncertainty and present bias. The closest related work is on the design of mechanisms to improve resource utilization where agents have uncertain future values [23], or to incentivize reliable demand-side response in electric power systems [21, 22]. The proposed welfare-optimal and reliable contingent payment mechanisms, however, no longer have dominant strategy equilibrium if agents are present-biased, in which case an agent’s expected utility may no longer be monotonically decreasing as the penalty increases. This work builds on Ma et al. [23], generalizing the model to incorporate present bias, and makes use of two-bid penalty bidding to align incentives for agents with any level of present bias or sophistication. Crucially, the proposed mechanism does not need to assume any knowledge about agents’ level of bias or value distributions.

As noted by Ma et al. [23], contingent payments have arisen in the past in the context of oil drilling license auctions [16], royalties [7, 10], ad auctions [28], and selling a firm [12]. Payments that are contingent on some observable world state also play the role of improving revenue as well as hedging risk [27]. In our model, in contrast, payments are contingent on agents’ own downstream decisions and serve the role of commitment devices. In regard to auctions in which actions take place after the time of contracting, Atakan and Ekmekci [2] study auctions where the value of taking each action depends on the collective actions by others, but these actions are taken before rather than after observing the world state. Courty and Li [9] study the problem of revenue maximization in selling airline tickets, where passengers have uncertainty about their value for a trip, and may decide not to take a trip after realizing their actual values. The type space considered there is effectively one-dimensional, and present bias is not considered.

Laibson [20] introduced quasi-hyperbolic discounting for modeling time-inconsistent decision making, where in addition to the standard exponential discounting, all future utilities are discounted by an additional present bias factor. Present-biased agents have been further classified into those who are sophisticated, and fully aware of their present bias and can anticipate their future self-control problem, and those who are naive, and blissfully unaware of their present bias and believe they will still have time-consistent behavior in the future [25]. O’Donoghue and Rabin [25] find through analysis that naivete is associated with procrastinating immediate-cost activities and doing immediate-reward activities too soon, while sophistication lessens the degree of procrastination but intensifies the doing-too-soon. O’Donoghue and Rabin [26] also study how the role of choice affects procrastination, and introduce the idea of a partially naive agent, who is aware of present bias but underestimates the degree of this bias. Researchers have also attempted to estimate the present bias factor in the real world based on observed behavior, however, there has not been consensus about the distribution of present bias in the population [3, 8, 13]

In a different setting, Kleinberg and Oren [18] consider how to modify the sequencing of tasks available to individuals in order to help a present-biased agent adopt a more optimal sequence of tasks. Kleinberg et al. [19, 17], Gravin et al. [15] later extended this work to consider sophisticated agents, the interaction between present bias and sunk-cost bias, and agents whose present bias factors are uncertain. There is no uncertain values or costs in these models, and no contention for limited resources. Other researchers have also examined the role of various kinds of commitment devices to mitigate present bias. Laibson [20] examines illiquid assets (that have long-term returns but cannot be realized immediately) as a self-control mechanism. Beshears et al. [5] also use liquidity as a commitment mechanism, testing how the design of contracts with varying levels of liquidity affects demand for commitment devices used to restrict spending. Giné et al. [14] test a savings account designed to serve as a commitment device to help with smoking cessation. By bundling a “want” activity (listening to one’s favorite audio book) with a “should” activity (going to the gym), Milkman et al. [24] evaluate the effectiveness of temptation bundling as a commitment device to tackle two self-control problems at a time.

2 Preliminaries

We first introduce the model for the assignment of homogeneous resources (leaving a discussion of the generalization to heterogeneous resources to Section 3.1). There is a set of agents and three time periods. In period , the value of each agent for using a resource is uncertain, represented by (the time line is more formally presented in the next subsection). The period 1 value from using the resource is a random variable with cumulative distribution function (CDF) , whose exact (and potentially negative) value is not realized until period . The period  value models the expected future benefit agent derives, if she uses a resource in period .

Agents are present-biased, that at any point of time, agent discounts her utility from all future periods by a factor of . The agents may not be fully aware of their bias, however, and agent believes that when making decisions, she will only discount her future utility by a factor of . An agent with is rational and does not discount her future utility. An agent with is said to be sophisticated, and is fully aware of the degree of her present bias. An agent with and is said to be naive, believing that she will make rational decisions in the future, and an agent with is said to be partially naive.

Let denote agent ’s type, and denote a type profile. The tuple is agent ’s private information at period , when the assignment of resources is determined. Each allocated agent decides on whether to use the resource at period , after she privately learns the realization of . Define . Following Ma et al. [23], we make the following assumptions about for each :

  1. , which means that a rational agent gets positive value from using the resource with non-zero probability, thus the option to use the resource has positive value.

  2. , which means that agents do not get infinite expected utility from the option to use the resource, thus would not be willing to pay an unboundedly large payment for it.

  3. , meaning that being forced to always use the resource regardless of what happens is not favorable for any agent, so that no agent would accept any unboundedly large no-show penalty for the right to use a resource.999 Regardless of the degree of present bias or sophistication, an agent for which (A3) is violated is willing to accept a 1 billion dollar no-show penalty, (almost) always use the resource, and get a non-negative utility in expectation.

We now provide a few examples of different models for agent types.

Example 1 ( model).

The future value for agent for using a resource is , however, she is able to do so only with probability and at a period  opportunity cost modeled by . With probability , agent is unable to show up to use the resource. This hard constraint can be modeled as taking value with probability . See Figure 2. We have , and thus (A1)-(A3) are satisfied.

some text

Figure 2: Distribution of under the type model.
Example 2 (Exponential model).

The opportunity cost for an agent to use the resource in period one is an exponentially distributed random variable with parameter , (i.e. ). If the agent used a resource, she gains a future utility of . See Figure 3. The expectation of is , thus and (A1)-(A3) are satisfied when .

some text

Figure 3: Agent period 1 value distribution under the exponential type model.

2.1 Two-Period Mechanisms

We consider two-period mechanisms defined by , following the timeline proposed in Ma et al. [23]. At period 0, each agent makes a report from some set of messages . Let denote a report profile. Based on the reports, an allocation rule assigns the right to use the resources to a subset of at most agents, namely those agents for whom . for all . Each agent is charged in period 1, and the mechanism also determines the penalty for each allocated agent for no-shows (we set for ).

The timeline of a two-period mechanism is as follows:

Period :

  1. Each agent reports to the mechanism based on the knowledge of .

  2. The mechanism allocates the resource to a subset of agents with , thus for all and for all .

  3. The mechanism determines the base payment for all agents , and the penalty for each allocated agent .

Period :

  1. The mechanism collects from each agent.

  2. Each allocated agent privately observes the realized values of , and decides on whether to use the resource based on , and .

  3. The mechanism collects the penalty from each allocated agent if she did not use the resource.

Example 3 ( price auction).

The standard price auction for assigning resources can be described as a two-period mechanism, where the report space is . Ordering agents in decreasing order of their reports, s.t. (breaking ties randomly), the allocation rule is for all , for . Each allocated agent is charged , and all other payments are zero. The price auction does not make use of any penalties.

Example 4 (Generalized contingent second price mechanism).

The generalized contingent second price (GCSP) mechanism [23] for assigning homogeneous resources collects a single bid from each agent, allocates the right to use resource to the highest bidders, and charges the highest bid, but only if an allocated agent fails to use the resource. Formally, . Ordering the agents s.t. (breaking ties randomly), we have for , for , , and all other payments are 0.

We assume that agents are risk-neutral, expected-utility maximizers with quasi-linear utility functions and quasi-hyperbolic discounting for future utilities.

An agent who is allocated a resource faces a two part payment , where is the period 1 penalty the agent pays in the case of no-show, and is the period 1 base payment the agent always pays. When period  arrives and the agent learns value (the realization of ), the agent discounts the future by , and believes that her utility from using the resource is . The utility from not using the resource is . Based on this, the agent uses the resource if and only if

(1)

breaking ties in favor of using the resource. Let be the indicator function, and define , the expected utility of the agent when facing penalty , as

(2)

The actual expected utility of an allocated agent facing two-part payment is . Under a two-period mechanism , and given report profile , agent ’s expected utility is .

Agents believe that they will make decisions as if they have present-biased factor , and will decide to use the resource in period 1 if and only if

(3)

Therefore, an agent believes that when facing a penalty , her expected utility will be

(4)

We call the subjective expected utility function of the agent. For sophisticated agents who are able to perfectly predict their future decisions (i.e. ), and coincide.

Throughout the paper, we assume that if allocated, agents’ decisions in period 1 are influenced by their present bias, but are otherwise rational. The interesting question is to study an agents’ incentives regarding reports in period , which are made based on , i.e., based on an agent’s belief about her expected utility. For any vector and any , we denote .

Definition 1 (Dominant strategy equilibrium).

A two-period mechanism has a dominant strategy equilibrium (DSE) if for each agent , for any type satisfying (A1)-(A3), there exists a report such that ,

Let denote the report profile under a DSE given type profile .

Definition 2 (Voluntary participation).

A two-period mechanism satisfies voluntary participation (VP) if for each agent , for any type satisfying (A1)-(A3), and any report profile ,

Voluntary participation requires that each agent believes that she has non-negative expected utility under her dominant strategy, given that she makes present-biased but otherwise rational decisions in period (if allocated), regardless of the reports made by the rest of the agents. Voluntary participation allows an agent to have negative utility at the end of period 1. We cannot charge unallocated agents without violating VP, thus for all , for all report profiles .

The expected revenue of a two-period mechanism is the total expected payment made by the agents to the mechanism in the DSE, assuming present-biased but otherwise rational decisions in period :

(5)
Definition 3 (No deficit).

A two-period mechanism satisfies no deficit (ND) if, for any type profile that satisfies (A1)-(A3), the expected revenue is non-negative: .

The utilization achieved by mechanism in the DSE is the expected number of used resources, which is equal to the summation of the probability with which each assigned agent uses the resource:

(6)

The expected social welfare achieved by mechanism is the total expected value derived by agents from using the resources:

(7)

Our objective is to design mechanisms that maximize expected social welfare. We do not consider monetary transfers as part of the social welfare function. The reason appears is that it affects the decision of the allocated agents in period .

3 The Two-Bid Penalty Bidding Mechanism

In this section, we introduce the two-bid penalty bidding mechanism, and prove that agents with , exponential, or uniform model types have simple dominant strategies under the two-bid penalty bidding mechanism.

Definition 4 (Two-bid penalty bidding mechanism).

The two-bid penalty bidding mechanism collects bids from agents in period 0, and reorders agents in decreasing order of s.t. (breaking ties randomly).

  1. Allocation rule: for , for .

  2. Payment rule: the mechanism announces , elicits a second bid from each assigned agent , and sets . for all , and for all .

The two-bid penalty bidding mechanism first asks agents to bid on the maximum penalties they are willing to accept for the option to use the resource for free, and assigns the resources to the highest bidders. The mechanism then asks each assigned agent to bid a penalty that is weakly higher than the bid, which is the amount she would like the mechanism to charge her in case of a no-show.

We first prove some useful properties of agents’ subjective expected utility function .101010Lemma 3.2 in Ma et al. [23] proved that for a rational agent without present bias, her expected utility as a function of the penalty is continuous, convex, and monotonically decreasing with . These properties no longer hold for agents with present bias.

\thmt@toks\thmt@toks

Given any agent with type that satisfies (A1)-(A3), the agent’s subjective expected utility as a function of the penalty satisfies:

  1. , .

  2. is right continuous, i.e. for all . Moreover, is upper semi-continuous, meaning that for all ,

\thmt@toks
\thmt@toks
Lemma 1.
Proof.

We first prove part (i). It is obvious that given (4). For the limit as , observe that can be rewritten as:

(8)

By the monotone convergence theorem, the first term of (8) converged to as . The second term is upper bounded by , therefore holds.

For part (ii), is a continuous function in , therefore its expectation is also continuous in . is right continuous, implying the right continuity of as well. The fact that and that only jumps up when it’s not continuous guarantees for all . ∎

For any penalty , we now define as the optimal, expected utility that agent believes she has, if agent can choose to be charged any penalty that is at least , i.e.

(9)

The following lemma makes use of Lemma 3, and proves the continuity, monotonicity, and the existence of a zero-crossing for — the maximum penalty an agent is willing to accept, if the agent can choose to be charged any penalty that is weakly above this amount.

\thmt@toks\thmt@toks

Given any agent with type that satisfies (A1)-(A3), the agent’s subjective expected utility as a function of the minimum penalty satisfies:

  1. is continuous and monotonically decreasing in .

  2. There exists a zero-crossing s.t. and for all .

\thmt@toks
\thmt@toks
Lemma 2.
Proof.

For part (i), the monotonicity of is obvious, and the continuity of is implied by the right continuity of . For part (ii), observe that since implies that there exists s.t. for all . The continuity of then implies that defining

we must have and for all . ∎

The following example illustrates the expected utility functions of an agent with type (see Example 1), and shows that there may not exist a dominant strategy under the CSP mechanism.

Example 5.

Consider a sophisticated agent whose type follow the model, who is assigned a resource and charged a no-show penalty . With probability , the agent is not able to use the resource at all and pays the penalty . With probability , the agent is able to use the resource at a cost of , but will use the resource if and only if . When , the agent’s expected utility as a function of the no-show penalty is of the form:

and we know since the agent is sophisticated. See Figure 3(a). Intuitively, is the minimum penalty the agent needs to be charged, so that she will show up to use the resource when she is able to. When , the agent ends up always paying the penalty, which is too small to incentivize utilization. of this agent is as shown in Figure 3(b). The maximum penalty the agent is willing to accept is .

(a) .

(b) .
Figure 4: Expected utility functions of a sophisticated agent with type, with .

There is no dominant strategy for this agent under the CSP mechanism. Consider the allocation of a single resource. If the highest bid among the rest of the agents satisfies , the agent gets positive utility from bidding , getting allocated and charged as penalty. However, if , bidding results in negative utility— the agent will be allocated, but and charged a penalty that is too small to overcome her present bias. In this case, the agent is better off bidding and not getting allocated. ∎

We now state and prove the main theorem of this paper.

\thmt@toks\thmt@toks

Given (A1)-(A3), under the two-bid penalty bidding mechanism, it is a dominant strategy for each agent to bid . If agent assigned a resource and given a minimum penalty , it is then a dominant strategy to bid . Moreover, the mechanism satisfies voluntary participation and no deficit.

Theorem 1 (Dominant strategy equilibrium of the two-bid penalty bidding mechanism).
Proof.

We first consider an agent who is assigned a resource, and asked the mechanism to bid an amount that is at least . The highest expected utility when the agent can choose any penalty weakly higher than is achieved at some because of the right continuity of . Since whichever amount an agent bids as will be the penalty she is charged by the mechanism, it is a dominant strategy to bid .

Given that an assigned agents will get expected utility when she is asked to bid a penalty at least , is effectively her expected utility function in the first round of bidding. With the monotonicity of and the fact that the minimum penalty is determined by the highest bid, it is standard that an agent bids in DSE the highest “minimum penalty to choose from” that she is willing to accept, which is . ∎

Example 6.

Consider the allocation of one resource to two agents with types, where:

  1. , , , ,

  2. , , , .

When , agent never uses the resource. On the other hand, means that agent  uses the resource with probability given any non-negative penalty. for since both agents are sophisticated, and the expected utility functions of the two agents are as shown in Figure 5.

Figure 5: Expected utility functions of two agents in Example 6.

Under the second price auction, agents will bid in DSE and — the value of the option to use the resource without any no-show penalty. Knowing that she will never show up without penalty, agent is not willing to pay any positive amount for the option to use the resource. Agent gets assigned the resource and charged no penalty, achieving social welfare and utilization .

Under the two-bid penalty bidding mechanism, the agents bid in DSE , and . Agent  is therefore assigned and will bid when asked to choose a penalty weakly above , since is monotonically decreasing in for . As a result, the two-bid penalty bidding achieves social welfare and utilization — both are higher than those under the second price auction. ∎

3.1 Discussion

For rational agents with , the subjective expected utility as a function of the penalty is monotonically decreasing, therefore and coincide. In this case, the equilibrium outcome under the two-bid penalty bidding mechanism coincides with that under the -price generalization of the CSP mechanism.

Since is what an agent considers during the bidding process, in period 0 a naive agent behaves as if she was rational with the same value distribution. In period 1, however, present bias will take effect, and the naive agent may make sub-optimal decisions. The actual expected utility a naive agent gets from participating in the mechanism, therefore, may be negative, despite the fact that she is willing to participate, and that she believes she will get non-negative expected utility.

Taking the derivative of the subjective expected utility (4) w.r.t. , we can show that for two agents and who are identical except that , for all . As a result, . This implies that an agent who believes that she is less present-biased will bid higher under both the two-bid penalty bidding mechanism and the price auction.

The CSP mechanism is provably utilization-optimal among a large family of mechanisms with a set of desirable properties [23]. The two-bid penalty bidding mechanism, however, does not optimize utilization— the reason is that the actual present bias factor does not affect a naive agent’s bid, and thus it is still possible for a very biased naive agent to be assigned under the two-bid penalty bidding mechanism but never show up. On the other hand, the auction may not assign the resource to this agent, and for this reason may achieve higher utilization and welfare.

For assigning multiple heterogeneous resources, we can consider a model with resources , where each agent has a random value for using resource . In this case, and can be defined similarly to (4) and (9). The two-bid mechanism can be generalized through the use of a minimum Walrasian equilibrium price mechanism that computes the assignment, and the minimum penalty each agent faces using  [11, 1, 23]. As a second step, each assigned agent is then are asked to report a weakly higher penalty that she wants to be charged by the mechanism. The same DSE analysis then holds.

4 Simulation Results

In this section, we adopt the exponential type model introduced in Example 2, and compare in simulation the social welfare and utilization achieved by different mechanisms and benchmarks for assigning five homogeneous resources. Additional simulations for the exponential type model are presented in Appendix A, together with similar results when assuming the type model (see Example 1) or a uniform type model where agents’ period 1 values are uniformly distributed.

Recall that in the exponential model, , thus where is the expected period 1 opportunity cost for using the resource. Agents’ expected utility functions and DSE bids are provided in Appendix B. We consider a type distribution in the population, where the value and the expected opportunity cost for each agent are both uniformly distributed:

With with probability 1, assumptions (A1)-(A3) are satisfied. The simulation results are not sensitive to the choices of parameter , and we fix for the rest of this section.

4.1 Varying Resource Scarcity

Fixing the number of resources at , we study the impact of varying the scarcity of the resource, by examining the outcome under different mechanisms and benchmarks as the number of agents varies from to . The “First Best” benchmark is the highest utilization or welfare that is achievable without violating voluntary participation or no deficit. The “first-come-first-serve” (FCFS) benchmark assumes a random order of arrival, effectively assigning to a random subset of agents, and does not charge any base payment or penalty.

Naive Agents

We first consider the scenario where all agents are naive and unaware of their present bias. The present bias factor is assumed to distribute uniformly on , and all agents believe that they have . As the number of agents varies from to , we compute the average social welfare and utilization over 10,000 randomly generated profiles under the different mechanism and benchmarks, as shown in Figure 6.

(a) Social Welfare.
(b) Utilization.
Figure 6: Social welfare and utilization for naive agents with exponential types.

When the number of agents is small, the outcomes under the two-bid penalty mechanism, the price auction, and first-come-first-serve are similar, since all three mechanisms effectively assign the resources to all agents, without charging any penalty. As the number of agents increases, the two-bid penalty bidding achieves higher social welfare and significantly higher utilization than the price auction (which is welfare optimal for rational agents without present bias), and does this without charging any payments from agents who actually show up. Both mechanisms significantly outperform the first-come-first-serve benchmark, which is analogous to the reservation system widely used in practice.

Sophisticated Agents

We now consider settings with fully sophisticated agents, whose present-biased factor are distributed as , but for all . We vary the number of agents from to , and show the average social welfare and utilization over 10,000 randomly generated economies in Figure 7.

(a) Social Welfare.
(b) Utilization.
Figure 7: Social welfare and utilization for sophisticated agents with exponential types.

Similar to the setting with naive agents, the two-bid penalty bidding mechanism achieves higher welfare and utilization than the price auction. These gains are slightly smaller than in the setting with naive agents, moreover, the price auction achieves relatively higher gain in utilization when compared to the first-come-first-serve baseline. This improved performance of the price auction in the case of sophisticates comes out because sophisticated agents are able to adjust their bids depending on their present bias level, and avoid the situation where a naive agent bids too much, gets assigned, but rarely show up, resulting in very low utilization, welfare, and negative actual expected utility for the agent herself.

In Appendix A.1, we provide additional simulation results for exponential types as varies, for the settings where all agents are fully rational (), and where all agents are partially naive (in which case we assume ). The outcome for partially naive agents is between the outcome for fully naive agents and fully sophisticated agents. For fully rational agents, the two-bid penalty bidding mechanism achieves slightly worse welfare than the price auction, which is provably optimal for this setting. Two-bid penalty bidding, however, still achieves higher utilization and also a significantly better outcome than the FCFS benchmark.

4.2 Impact on Agents with Different Degrees of Bias

In this section, we consider a population of agents with varying levels of present bias, and study the different outcomes for agents with different degrees of present bias under various mechanisms and benchmarks. We consider the same distribution of the expected opportunity cost and the future value as in the earlier setting, but fix the total number of agents at , and also fix the preset bias factor of agent at for all 1,000,000 randomly generated economies. The smaller an agent’s index is, the more present-biased the agent.

Naive Agents

We first consider the scenario where all agents are naive. The average (per economy) welfare and utilization of each agent is as shown in Figure 8. Under the first-best welfare and the first-best utilization, agents with different degrees of bias achieve the same welfare and utilization. This is because the agents all have the same distribution of and , and only differ in their bias factor . The full-information first best knows types of agents, and adjusts the penalties accordingly, so that there is no difference between agents who are more or less biased. Also note that naive agents behave in period 0 as if they are rational, therefore all agents bid in the same way despite their degree of present bias, and are allocated with the same average probability.

(a) Social Welfare.
(b) Utilization.
Figure 8: Welfare and utilization for naive agents with exponential types, fixing .

From Figure 7(b), we see that all agents achieve higher utilization under two-bid penalty bidding than under the price auction. Agents who are less biased (higher indices) achieve higher utilization than those who are more biased (lower indices), but agents who are more biased achieve a higher gain in two-bid penalty bidding compared with the price auction. Perhaps more importantly, Figure 7(a) shows that the less biased agents obtain much higher welfare under the price auction. The two-bid penalty bidding mechanism, in comparison, helps the agents who are more biased to achieve significantly higher social welfare, and at the same time results in a slight lower welfare for the least-biased agents: this decrease is because for the least biased agents, her own period  decision will be close to optimal, and charging a penalty may sometimes incentivize her to use the resource even when the actual future value is smaller than present cost.

Overall, the outcome under the two-bid penalty bidding mechanism is significantly more equitable for agents with all levels of bias. It is also worth noting that while naive agents do not see the value of commitment and generally do not take any commitment device when offered [6, 4], the two-bid penalty bidding mechanism is able to help the naive agents, since a commitment device is designed through the mechanism, and not accepting a commitment device is not an option.

(a) Social Welfare.
(b) Utilization.
Figure 9: Welfare and utilization for sophisticated agents with exponential types, fixing .
Sophisticated Agents

We now consider the setting where all agents are sophisticated. The average welfare and utilization of each agent is as shown in Figure 9. The first observation is that under the price auction, the welfare and utilization for the most biased agents is effectively zero, while the least biased agents achieve better welfare and utilization than the first-best outcome. This is because when the bids of sophisticated agents factor in the level of present bias, the more biased agents bid lower than the less biased agents, and therefore get assigned with lower probability. The more biased sophisticated agents bid lower under two-bid penalty bidding as well, and as a result the two-bid penalty bidding is not able to achieve the same level of social welfare for all agents. Nevertheless, it achieves large improvements for the most biased part of the population.

5 Conclusion

We propose the two-bid penalty-bidding mechanism for resource allocation in the presence of uncertainty and present bias. We prove the existence of a simple dominant strategy equilibrium regardless of an agent’s value distribution, level of present bias, or degree of sophistication. Simulation results show that the proposed mechanism not only improves utilization, but also achieves higher welfare than mechanisms that are welfare-optimal for settings without present bias.

Ongoing work includes empirical work in collaboration with exercise studios and event organizers in order to better understand people’s behavior, with the goal of seeking to separate the impact of uncertainty and present bias on utilization. Another interesting direction for future work is to generalize the model to allow for more than two time periods, where agents may arrive asynchronously, when uncertainty unfolds gradually over time, and where resources can be re-allocated.

References

  • Alaei et al. [2016] S. Alaei, K. Jain, and A. Malekian. Competitive equilibria in two-sided matching markets with general utility functions. Operations Research, 64(3):638–645, 2016.
  • Atakan and Ekmekci [2014] Alp E Atakan and Mehmet Ekmekci. Auctions, actions, and the failure of information aggregation. American Economic Review, 104(7), 2014.
  • Augenblick and Rabin [2015] Ned Augenblick and Matthew Rabin. An experiment on time preference and misprediction in unpleasant tasks. Review of Economic Studies. Forthcoming, 2015.
  • Beshears et al. [2011] John Beshears, James J Choi, David Laibson, Brigitte C Madrian, and Jung Sakong. Self control and liquidity: How to design a commitment contract. 2011.
  • Beshears et al. [2015] John Beshears, James J Choi, Christopher Harris, David Laibson, Brigitte C Madrian, and Jung Sakong. Self control and commitment: Can decreasing the liquidity of a savings account increase deposits? Working Paper 21474, National Bureau of Economic Research, August 2015. URL http://www.nber.org/papers/w21474.
  • Bryan et al. [2010] Gharad Bryan, Dean Karlan, and Scott Nelson. Commitment devices. Annu. Rev. Econ., 2(1):671–698, 2010.
  • Caves [2003] Richard E Caves. Contracts between art and commerce. Journal of economic Perspectives, pages 73–84, 2003.
  • Cohen et al. [2016] Jonathan D Cohen, Keith Marzilli Ericson, David Laibson, and John Myles White. Measuring time preferences. Working Paper 22455, National Bureau of Economic Research, July 2016. URL http://www.nber.org/papers/w22455.
  • Courty and Li [2000] Pascal Courty and Hao Li. Sequential screening. The Review of Economic Studies, 67(4):697–717, 2000.
  • Deb and Mishra [2014] Rahul Deb and Debasis Mishra. Implementation with contingent contracts. Econometrica, 82(6):2371–2393, 2014.
  • Demange and Gale [1985] G. Demange and D. Gale. The strategy structure of two-sided matching markets. Econometrica, 53:873–888, 1985.
  • Ekmekci et al. [2016] Mehmet Ekmekci, Nenad Kos, and Rakesh Vohra. Just enough or all: Selling a firm. American Economic Journal: Microeconomics, 8(3):223–56, 2016.
  • Ericson and Laibson [2018] Keith Marzilli Ericson and David Laibson. Intertemporal choice. Working Paper 25358, National Bureau of Economic Research, December 2018. URL http://www.nber.org/papers/w25358.
  • Giné et al. [2010] Xavier Giné, Dean Karlan, and Jonathan Zinman. Put your money where your butt is: A commitment contract for smoking cessation. American Economic Journal: Applied Economics, 2(4):213–35, October 2010. doi: 10.1257/app.2.4.213. URL http://www.aeaweb.org/articles?id=10.1257/app.2.4.213.
  • Gravin et al. [2016] Nick Gravin, Nicole Immorlica, Brendan Lucier, and Emmanouil Pountourakis. Procrastination with variable present bias. In Proceedings of the 2016 ACM Conference on Economics and Computation, pages 361–361. ACM, 2016.
  • Hendricks and Porter [1988] Kenneth Hendricks and Robert H Porter. An empirical study of an auction with asymmetric information. The American Economic Review, pages 865–883, 1988.
  • Kleinberg et al. [2017] Jon Kleinberg, Sigal Oren, and Manish Raghavan. Planning with multiple biases. In Proceedings of the 2017 ACM Conference on Economics and Computation, pages 567–584. ACM, 2017.
  • Kleinberg and Oren [2014] Jon M. Kleinberg and Sigal Oren. Time-inconsistent planning: A computational problem in behavioral economics. CoRR, abs/1405.1254, 2014. URL http://arxiv.org/abs/1405.1254.
  • Kleinberg et al. [2016] Jon M. Kleinberg, Sigal Oren, and Manish Raghavan. Planning problems for sophisticated agents with present bias. CoRR, abs/1603.08177, 2016. URL http://arxiv.org/abs/1603.08177.
  • Laibson [1997] David Laibson. Golden eggs and hyperbolic discounting. The Quarterly Journal of Economics, 112(2):443–478, 1997.
  • Ma et al. [2016] Hongyao Ma, Valentin Robu, Na Li, and David C. Parkes. Incentivizing reliability in demand-side response. In Proceedings of The 25th International Joint Conference on Artificial Intelligence (IJCAI’16), pages 352–358, 2016.
  • Ma et al. [2017] Hongyao Ma, Valentin Robu, and David C. Parkes. Generalizing Demand Response Through Reward Bidding. In Procedings of the 16th International Conference on Autonomous Agents and Multiagent Systems (AAMAS’17), pages 60–68, 2017.
  • Ma et al. [2019] Hongyao Ma, Reshef Meir, David C. Parkes, and James Zou. Contingent payment mechanisms for resource utilization. In Proceedings of the 18th Conference on Autonomous Agents and MultiAgent Systems (AAMAS’19). arXiv preprint arXiv:1607.06511. IFAAMAS, 2019.
  • Milkman et al. [2013] Katherine L Milkman, Julia A Minson, and Kevin GM Volpp. Holding the hunger games hostage at the gym: An evaluation of temptation bundling. Management science, 60(2):283–299, 2013.
  • O’Donoghue and Rabin [1999] Ted O’Donoghue and Matthew Rabin. Doing it now or later. American Economic Review, 89(1):103–124, 1999.
  • O’Donoghue and Rabin [2001] Ted O’Donoghue and Matthew Rabin. Choice and procrastination. The Quarterly Journal of Economics, 116(1):121–160, 2001.
  • Skrzypacz [2013] Andrzej Skrzypacz. Auctions with contingent payments—an overview. International Journal of Industrial Organization, 31(5):666–675, 2013.
  • Varian [2007] Hal R Varian. Position auctions. international Journal of industrial Organization, 25(6):1163–1178, 2007.

Appendix

Appendix A Additional Simulation Results

a.1 Additional Results for Exponential Model

We first consider the identical setup as analyzed in Section 4, where agents have exponential types, and there are homogeneous resources to assign. We present the results as the number of agents varies, for settings where agents are all fully rational, or where they are all partially naive.

Fully Rational Agents

Figure 10 presents the average welfare and utilization of 10,000 randomly generated economies, assuming all agents are fully rational with . The two-bid penalty bidding mechanism achieves slightly worse welfare than the price auction, which is provably optimal for this setting. Two-bid penalty bidding, however, still achieves higher utilization and also significantly better outcome in comparison to FCFS.

(a) Social Welfare.
(b) Utilization.
Figure 10: Social welfare and utilization for rational agents with exponential types.
Partially Naive Agents

We now consider partially naive agents, where each agent has , and . The average welfare and utilization over 10,000 random economies are as shown in Figure 11. The outcome is in between the fully sophisticated and the fully naive settings discussed in the body of the paper.

(a) Social Welfare.
(b) Utilization.
Figure 11: Social welfare and utilization for partially naive agents with exponential types.

a.2 The Type Model

In this section, we compare the performance of different mechanisms and benchmarks for the type model introduced in Example 1. Agents’ expected utility functions and DSE bids are provided in Appendix B. We consider a type distribution in the population, where the value , cost , and reliability are all uniformly distributed:

With and with probability 1, assumptions (A1)-(A3) are satisfied almost surely. The results are not sensitive to the choices of parameter , and we fix for all results presented in the rest of this section.

a.2.1 Varying Resource Scarcity

Fixing the number of resources at , we first examine the outcomes as the number of agents varies from to . When all agents are naive with and . The average social welfare and utilization over 10,000 randomly generated profiles are as shown in Figure 12. For economies with fully sophisticated agents, where present-biased factors are distributed as , but for all , the social welfare and utilization average over 10,000 randomly generated economies are as shown in Figure 13.

(a) Social Welfare.
(b) Utilization.
Figure 12: Social welfare and utilization for naive agents with types.
(a) Social Welfare.
(b) Utilization.
Figure 13: Social welfare and utilization for sophisticated agents with types.

a.2.2 Impact on Agents with Different Degrees of Bias

We now consider a population of agents with the same distribution of , and as in the previous setting, but where the total number of agents is fixed at , and the present bias factor of agent each agent is fixed at . Assuming all agents are naive, the average welfare and utilization of each agent (over 1,000,000 randomly generated economies) is as shown in Figure 14. Assuming that agents are fully sophisticated instead, the average welfare and utilization of each agent is as shown in Figure 15.

(a) Social Welfare.
(b) Utilization.
Figure 14: Welfare and utilization for naive agents with types, fixing .
(a) Social Welfare.
(b) Utilization.
Figure 15: Welfare and utilization for sophisticated agents with types, fixing .

a.3 Uniform Type Mode

We now consider agents with period  values uniformly distributed as in the following Example 7.

Example 7 (Uniform model).

In period , each agent incurs a uniformly distributed opportunity cost for using the resource, i.e. . If the agent used a resource, she gains a expected future utility of . See Figure 16. , thus and (A1)-(A3) are satisfied as long as .

some text

Figure 16: Agent period 1 value distribution under the uniform type model.

Agents’ expected utility functions and DSE bids under the uniform type model are derived in Appendix B. We consider a type distribution in the population, where and , are both uniformly distributed:

With with probability 1, assumptions (A1)-(A3) are satisfied almost surely. The results are not sensitive to the choices of parameter , and we fix for all results presented in the rest of this section.

a.3.1 Varying Resourc