TimeInconsistent Bargaining and CrossCommitments
1. Introduction
While there is a growing literature on the effects of presentbiased preferences on bargaining decisions, commentators have not yet given sufficient attention to a number of factors that will determine the extent to which players will incur large welfare losses due to their presentbiased preferences. These factors include the relative timing of bargaining costs and bargaining rewards; the likelihood of repeated bargaining procrastination, preconsumption, and overconsumption; the interplay between selfawareness and counterpartyawareness; and the efficient use of selfcommitment and crosscommitment devices.
This paper sets forth a general model of timeinconsistent bargaining in which bargaining payoffs can occur in the same or in different periods and in which players have different levels of awareness of their own and the other player’s presentbiased preferences. Moreover, the players can create crosscommitment devices by determining the relative timing of bargaining payoffs. The paper shows that presentbiased preferences can affect bargaining outcomes only in contexts involving immediate and delayed bargaining payoffs. In particular, if players can reach a bargain in period 1 and experience all bargaining payoffs in period 2, they will behave in the same manner as the timeconsistent, exponential discounters of standard bargaining models. Moreover, in takeitorleaveit bargaining situations in which players experience all bargaining payoffs in the period in which they reach a bargain, the players will again behave in a timeconsistent manner. Players in the latter two types of bargaining scenarios will incur no welfare losses due to their presentbiased preferences.
The paper distinguishes between two types of timeinconsistent bargaining contexts. In the first, players incur immediate bargaining costs in order to produce delayed bargaining rewards. In immediatecosts scenarios, timeinconsistent players will have an incentive to procrastinate agreeing to a bargain and following through with other planned bargaining actions. In the second context, players receive immediate bargaining rewards and incur delayed bargaining costs. In immediaterewards scenarios, players will have an incentive to preconsume, agreeing to bargains too soon, and to overconsume, agreeing to one or more nonoptimal bargains solely due to the added weight that timeinconsistent players give to immediate rewards. The paper shows that even a relatively small preference for immediate gratification can lead to repeated bargaining procrastination, and thus to large aggregate welfare losses. Repeated procrastination can help explain observed bargaining delays that are sometimes blamed on other types of informational frictions. Overconsumption of inefficient bargains can also produce large aggregate welfare losses for players who enter into a large number of transactions. Preconsumption, on the other hand, can sometimes improve the players’ joint welfare since it can increase the likelihood that timeinconsistent players will follow through with a planned bargain. This will be true in bargaining scenarios in which the efficient result is for the players to agree to a bargain immediately. Drawing a clear distinction between bargaining procrastination, overconsumption, and preconsumption is therefore important for determining when presentbiased preferences can produce welfare losses.
The paper makes three main contributions to the literature on timeinconsistent bargaining. First, the paper shows that the relative timing of bargaining payoffs is a key driver of timeinconsistent bargaining. Moreover, the paper shows that drawing a clear distinction between procrastination, preconsumption, and overconsumption can help to better specify the contexts in which timeinconsistent bargaining will occur. The paper’s second contribution is to identify contexts in which crosscommitment and preplay commitment devices come into play and the consequences of using those devices. Relatedly, the paper shows that, in order to minimize the costs of crosscommitment and precommitment devices, players should first target immediate payoffs.
The paper’s third contribution is to identify a set of intertemporal bargaining decisions that are particularly susceptible to preference reversals. We show that, all other things being equal, timeinconsistent bargainers are more likely to: delay agreeing to a bargain; reject worthwhile bargains and accept nonworthwhile ones; and delay entering and exiting bargaining relationships, including delaying exercising their outside options. The current literature on timeinconsistent bargaining has focused on how presentbiased preferences affect the two key bargaining decisions: making a proposal and accepting or rejecting a proposal. We show that the same factors play a role in other bargaining decisions, such as entry and exit decisions.
2. TimeInconsistent Bargaining
We assume that two players bargain over the division of a bargaining surplus. We allow for takeitorleaveit offers as well as sequential bargaining involving offers and counteroffers. The bargaining surplus depends both on (1) the magnitude of positive and negative bargaining payoffs (bargaining “rewards” and “costs”, respectively) and (2) when those payoffs materialize. Players can make bargaining proposals that both allocate the net surplus and determine the timing of receiving bargaining rewards and incurring bargaining costs.
A longterm decision in period t is one in which a player compares two or more future payoffs and chooses the optimal course of action for period t + 1.
A shortterm decision in period t + 1 is one in which a player compares both immediate and future payoffs and determines whether to reverse a periodt longterm decision.
Players exhibit timeconsistent (“TC”) preferences if in period t they make a longterm decision to do A in period t + 1, and in period t + 1, they make a shortterm decision to do A.
Players exhibit timeinconsistent (“TI”) preferences if in period t they make a longterm decision to do A in period t + 1, and in period t + 1, they make a shortterm decision not to do A, and they do so solely due to their presentbiased preferences.
2.1. TimeInconsistent Bargaining Requires Existence of Tradeoffs between Immediate and Delayed Payoffs
U_{t} = (u_{t} + δu_{t+}_{1} + δ^{2}u_{t+}_{2} + δ^{3}u_{t+}_{3} +…δ^{n}u_{t+n}, …)
It can be easily checked that exponential discounters discount in the same manner in period t and all subsequent periods—i.e., they always discount a oneperiod delay by δ, a twoperiod delay by δ^{2}, and so on.
However, the evidence on presentbiased preferences suggests both that agents have declining discount rates and that they discount most heavily when making decisions involving immediate payoffs. For example, when asked to choose between USD 1 a year from today and USD 2 in a year and a day, people routinely chose the USD 2. However, when asked to choose between USD 1 today and USD 2 tomorrow, people choose the immediate USD 1. An exponential discounter might discount the oneday delay in both instances by the same discount factor δ.
U_{t} = (u_{t} + βδu_{t+1} + βδ^{2}u_{t+2} + βδ^{3}u_{t+3} + … βδ^{n}u_{t+n}, …)
Identical TC and TI players will act in the same manner in all periods involving only delayed payoffs. However, in periods involving immediate and delayed payoffs, TI players will give 1/β greater weight to the immediate payoffs than do TC players.
Suppose that both types of players are in period t − 1 considering the following delayed payoffs: TC: U_{t}_{−1} = (δu_{t} + δ^{2}u_{t+}_{1}+ δ^{3}u_{t+}_{2} +…δ^{n+}^{1}u_{t+n}, …) and TI: U_{t}_{−1} = (βδu_{t} + βδ^{2}u_{t+}_{1} + βδ^{3}u_{t+}_{2} +…βδ^{n+}^{1}u_{t+n}, …). Since β remains constant, one can divide it out and show that when payoffs are all delayed both the TC and TI players reach the same longterm decision. But in period t, the TC player still discounts period t + 1 payoffs by δ: U_{t} = (u_{t} + δu_{t+}_{1} + δ^{2}u_{t+}_{2} +…δ^{n}u_{t+}_{n}, …); while the TI player discounts period t + 1 payoffs more severely, by βδ: U_{t} = (u_{t} + βδu_{t+}_{1} + βδ^{2}u_{t+}_{2} +…βδ^{n}u_{t+n}, …). Dividing by β shows that the TI player gives 1/β greater weight to u_{t} over u_{t+}_{1} than does the TC player. Moreover, from the perspective of period t, both players discount delayed payoffs from period t + 1 onward identically. TC: U_{t} = (δu_{t+}_{1} + δ^{2}u_{t+}_{2} +…δ^{n}u_{t+n}, …); and TI: U_{t} = (βδu_{t+}_{1} + βδ^{2}u_{t+}_{2} +…βδ^{n}u_{t+n}, …). One can again divide out the constant β and show that when payoffs are all delayed both the TC and TI players reach the same decision. Since by assumption, both the TC and TI players are otherwise identical, it follows that the only time in which their behavior will differ is when they are making decisions involving a tradeoff between immediate and delayed payoffs.
Proposition 1 shows that not all bargaining contexts involving TI players will lead to TI bargaining behavior. In bargaining games in which players reach a bargain in period t and all payoffs materialize in period t + 1 or later, both TC and TI players will make the same longterm and shortterm bargaining decisions. Recall that a TI player with a β = 0.5 (i.e., an immediacy multiplier of 2) gives twice as much weight to immediate period t payoffs, u_{t}, as compared to delayed period t + 1 payoffs, u_{t+}_{1}. However, when players reach a bargain in period t and all bargaining payoffs materialize in period t + 1, both TC and TI players experience the same instantaneous utility of 0 in period t. Moreover, both the TC and TI player discount the delayed period t + 1 payoffs by the same δ and reach the same decision.
2.2. Immediate Costs and Immediate Rewards of Bargains: Procrastination, Preconsumption, and Overconsumption
We must therefore distinguish between two types of TI bargaining scenarios. In the first type, TI players incur immediate costs to produce delayed bargaining rewards. This can lead to inefficient bargaining delays: bargaining procrastination.
In the second type of scenario, TI players receive immediate rewards and incur delayed bargaining costs. Under this second scenario, TI players may agree to bargains sooner than optimal and may agree to one or more bargains that TC players might reject: bargaining preconsumption and overconsumption, respectively.
A TI player preconsumes when she makes a longterm decision in period t to do A in period t + 2 given that doing so might maximize her intertemporal utility, but when period t + 1 arrives, she makes a shortterm decision to do A immediately solely due to the added weight that she gives to the immediate rewards of doing A. Relatedly, a TI player overconsumes when she makes a longterm decision in period t not to do A more than n times (where n may be 0) in periods t + 1 through T given that doing so might maximize her intertemporal utility, but she makes one or more shortterm decisions to do A more than n times solely due to the added weight that she gives to the immediate rewards of doing A. TC players will never preconsume or overconsume.
Suppose that a TC and a TI player each has a δ = 1, and the TI player, a β = 0.5. Suppose further that from the longterm perspective of period 0, both players decide to do an onerous bargaining task, A, in period 1. The task might require them to incur immediate costs—e.g., effort—with negative utility of 100 to produce delayed bargaining rewards of 150 in period 2. In period 1, the TC player does A, but the TI player procrastinates. She applies her immediacy multiplier, 1/β = 2, to the immediate costs of 100 and overrides her longterm decision, given that 200 > 150. Alternatively, suppose that the same two players can receive immediate bargaining rewards of 100 in period 1 and that partaking of the bargaining rewards will produce delayed bargaining costs of 150 in period 2. In period 0, both players decide to abstain from partaking of the period1 bargaining rewards. In period 1, the TC player abstains, but the TI player overconsumes, given that she now perceives the immediate rewards as 200 and the delayed costs as 150.
2.3. OneShot Bargaining Procrastination, Preconsumption, and Overconsumption and Maximum Welfare Loss
We have seen that TI players will reach different bargaining outcomes than their TC counterparts only in contexts in which bargaining payoffs materialize in different periods. We have also seen that the effects of a player’s presentbiased preferences on bargaining outcomes will differ depending on the relative timing of bargaining costs and rewards. But these two conditions by themselves are not sufficient for TI bargaining to occur. As we will now show, TI players will reverse their longterm bargaining decisions only when their presentbias, captured by their immediacy multiplier, is sufficiently great to lead them to conclude that the immediate gains from reversing their longterm decisions are greater than the delayed losses from doing so. We will also see that TI players who override their longterm decisions just once will have a maximum welfare loss that is bounded by their immediacy premium.
2.3.1. The Incentive to Override a LongTerm Optimal Plan
In period 1, players make shortterm decisions about planned actions and nonactions. Players may (1) follow through with their period0 plans, (2) override them, or (3) update them. TI players override their period0 plans whenever they conclude that the immediate gains from doing so exceed the delayed losses. Players who override their optimal period0 plans solely due to their presentbias incur welfare losses.
Period1 Immediate Gains of Overriding Optimal Plan—The Immediacy Premium:
The immediate gains depend on both the immediate payoffs and the TI player’s immediacy multiplier 1/β > 1. In the case of immediate costs, a player’s immediacy premium, (1/β × c_{t}) − c_{t}, captures her perceived immediate gains from delaying incurring immediate costs, c_{t}, until period 2. For example, if a player with a β = 0.75 and immediacy multiplier 1/β = 1.33 must incur immediate bargaining costs of 100, she perceives those costs as 133. She thus has an immediacy premium of (1.33 × 100) − 100 = 33. In the case of immediate bargaining rewards of 100, the immediacy premium is again 33.
Period2 Delayed Losses of Overriding Optimal Plan:
2.3.2. Welfare Losses from TI Bargaining Behavior
2.3.3. Bounded Welfare Loss in OneShot Preference Reversals
The maximum welfare loss from yielding to immediate gratification just once is bounded by a TI player’s immediacy premium. The TI player in Example 1 has an immediacy premium of 33 and will procrastinate, preconsume, or overconsume only if the delayed losses from doing so are less than 33 (assuming the player adheres to her longterm decision when indifferent). On the other hand, repeated timeinconsistent behavior can lead to very large aggregate welfare losses. Moreover, even a relatively small immediacy premium can lead to repeated procrastination, preconsumption, and overconsumption.
Suppose that a TI player has a relatively small presentbias captured by an immediacy multiplier 1/β = 1.11 (and without loss of generality, she has no longterm impatience: δ = 1). Suppose that every day she can choose to either exercise an outside option—at an immediate cost of 100—or delay until the following day, where each oneday delay reduces the option’s future rewards by 10. Moreover, assume that the player has made a longterm decision that exercising the outside option immediately will maximize her net bargaining surplus. Nonetheless, each day the TI player has an incentive to procrastinate: the delayed loss of 10 is less than the immediacy premium of 11 = (1.1 × 100) − 100.
2.4. Conditions for Repeated Preference Reversals and Large Welfare Losses
3. SelfAwareness and SelfCommitment
3.1. Naïve TI Players
Naïve TI players have a β < 1 and thus an immediacy multiplier 1/β > 1. However, in each period, they believe incorrectly that in the following period their future selves will exhibit a β = 1 and act in a TC fashion. They reach this conclusion even while they are yielding to temptation in the current period. Naïve players will never see the need to adopt commitment devices.
Thus, naïve players may repeatedly procrastinate agreeing to worthwhile bargains, ending fruitless bargaining relationships, and exercising outside options. Repeated procrastination by naïve TI players may explain, at least in part, observed inefficient delays in bargaining. Additionally, naïve bargainers faced with the prospect of immediate bargaining rewards may repeatedly preconsume and overconsume. For example, a naïve TI player in a longterm contract may engage in nibbling opportunism, repeatedly grabbing small immediate rewards—e.g., cutting corners and shirking—each time thinking incorrectly that it is the last time. Over time, repeated nibbling opportunism can lead to contract breaches and erosion of reputation.
3.2. Sophisticated TI Players
Sophisticated TI players correctly predict their future immediacy multiplier 1/β > 1 and immediacy premium. They know that their future selves will override optimal longterm bargaining decisions whenever the immediacy premium from doing so exceeds the delayed losses. They also have rational expectations, which will help them in cases of potential procrastination, but may hurt them in cases of potential preconsumption.
Suppose a sophisticated player must complete an onerous bargaining task in periods 1, 2, 3, or 4, and has made a longterm decision to complete the task in period 1. However, the sophisticated player also knows that each period she will compare an immediacy premium of 15 with a delayed procrastination loss of 10 and will have an incentive to procrastinate. She has rational expectations and thus knows that if she procrastinates until period 3, she will procrastinate until period 4. This means that if she procrastinates in period 2, she will end up procrastinating until period 4. Procrastinating for two periods produces an aggregate welfare loss of 20, which exceeds the period 2 immediacy premium of 15. Knowing that she will definitely complete the task in period 2, the sophisticated player concludes that she can safely procrastinate in period 1. The sophisticated player will suffer a relatively small welfare loss of 10 from procrastinating just once.
Suppose instead that a sophisticated player is in a bargaining game in which she will receive an immediate reward at the time of the bargain. Suppose also that she can agree to a bargain in periods 1, 2, 3, or 4, and that she has made a longterm decision that waiting until period 4 might maximize her bargaining surplus. Moreover, the sophisticated player knows that in each period she will compare an immediacy premium of 15 with a delayed loss of 10. The sophisticated player now reasons that if she reaches period 3, she will agree to the bargain then and not wait until period 4. She also reasons that if she reaches period 2, she will agree to the bargain then instead of waiting until period 3. As a result, she knows that she will have an incentive to agree to the bargain in period 1 given her immediacy premium of 15 and delayed loss of 10.
These two examples illustrate why it is important to draw a distinction between bargaining decisions involving immediate bargaining costs and those involving immediate bargaining rewards. In immediatecosts scenarios, sophistication helps, while in immediaterewards scenarios, sophistication hurts. Knowing this, a sophisticated player will have an incentive to engage in preplay commitment whenever she knows that she will be making shortterm bargaining decisions involving immediate rewards.
3.3. Partially Naïve TI Players
Partially naïve TI players know that their future selves will apply an immediacy multiplier 1/β > 1, but they underappreciate its true magnitude. For example, a partially naïve player may have a β = 0.5 and immediacy multiplier of 2 but believe incorrectly that she has a β = 0.9 and an immediacy multiplier of 1.1. Since it is the actual immediacy multiplier that will determine the immediacy premium, a partially naïve player who sufficiently underappreciates her true presentbias will act in the same manner as a fully naïve player, and one whose misprediction is less severe will act as a fully sophisticated player.
3.4. The Paucity of Learning
4. Counterparty Awareness and CrossCommitments
4.1. CounterpartySophisticated, CounterpartyNaïve, and CounterpartyPartiallyNaïve Players
TI players can both incur welfare losses and impose them on other players. As we will see below, these counterparty spillover welfare losses may exceed those incurred by the TI player. As a result, players in a bargaining game must try to predict the presentbiased preferences of other players. If player B has a β = 0.5 and player A believes incorrectly that B has a β > 0.5, player A’s decisions can lead to bargaining breakdowns or inefficient delays.
Let β_{CP} be player A’s beliefs of player B’s actual β.
Player A is counterpartysophisticated if player B has a β < 1 and player A correctly predicts it: β_{CP} = β.
Player A is counterpartynaive if player B has a β < 1 but player A believes incorrectly that B has TC preferences: β_{CP} = 1.
Player A is counterpartypartiallynaïve if she believes correctly that player B has a β < 1 but she mispredicts the true magnitude of player B’s presentbias: β < β_{CP} < 1.
Counterpartysophisticated players (and counterpartypartiallynaïve ones who are sufficiently sophisticated) will have an incentive to impose crosscommitment devices on their TI counterparties. Crosscommitments target either immediate or delayed payoffs. As we will see, all other things being equal, targeting immediate payoffs will produce the more efficient result. Additionally, crosscommitment devices can be used either benevolently to reduce welfare losses incurred by TI players or nonbenevolently to extract rents from TI players.
4.2. Welfare Losses Faced by CounterpartyNaïve Players
4.2.1. Ultimatum Game—Naïve TI Responder and CounterpartyNaïve Proposer
Suppose that TC player A is the proposer and player B the responder in an ultimatum game to divide π. Both players have δ = 1, but naïve player B has a β < 1. Player A is counterpartynaïve and thus believes incorrectly that she is bargaining with a TC player (i.e., β_{CP} = 1). Additionally, it is common knowledge that player B must incur immediate bargaining costs, c_{B1}, in period 1 to fully evaluate player A’s offer. If player B accepts the offer, he receives his portion of the surplus, v_{B2} (the delayed bargaining rewards) in period 2, and if he rejects, both players receive 0.
βv_{B2} − βc_{B1} ≥ 0 ≡ v_{B2} − c_{B1} ≥ 0 (given β is constant).
βv_{B2} − c_{B1} ≥ 0 ≡ v_{B2} ≥ c_{B1}/β
4.2.2. Spillover Welfare Losses Suffered by CounterpartyNaïve Player
Suppose that π = 1000, and v_{B2}, c_{B1} = 100, and that player B accepts the bargain when he is indifferent. Also suppose that player B has a β = 0.8. From a longterm perspective player B is indifferent, but from a shortterm perspective, he requires a delayed reward of at least: c_{B1}/β = 125.
Importantly, while it is player B’s timeinconsistency that leads him to reject the offer, he is made no worse off when he does so, since his share of the surplus is 0 whether he accepts or rejects. It is counterpartynaïve player A who suffers the full welfare loss from the bargaining breakdown.
4.2.3. Other Implications
4.3. Designing Optimal CrossCommitment Devices
In the previous section we showed that TI players can impose welfare losses on a counterpartynaïve player, even in cases in which the TI players suffer no welfare losses. A counterpartysophisticated player accurately predicts those welfare losses and may thus adopt crosscommitment devices to change the TI player’s incentives. Crosscommitment devices work by reallocating bargaining payoffs to offset the added weight that TI players give to immediate payoffs. This reallocation can be done either by targeting a TI player’s delayed payoffs or by targeting that player’s immediate payoffs. This section shows that targeting the TI player’s immediate payoffs always dominates targeting the player’s delayed payoffs. This result has important implications for the design of optimal crosscommitment devices.
4.3.1. CrossCommitment 1: Targeting Delayed Payoffs
4.3.2. CrossCommitment 2: Targeting Immediate Payoffs
Player A can design a crosscommitment device that targets B’s immediate payoffs by either increasing B’s immediate bargaining rewards and/or reducing his immediate bargaining costs. It is easy to see that crosscommitments that target B’s delayed payoffs are always more costly to implement than those that target immediate payoffs.
All other things being equal, crosscommitment devices that target a TI player’s delayed payoffs are always dominated by crosscommitments that target the TI player’s immediate payoffs.
To see this, assume without loss of generality that δ = 1. A TI player perceives an immediate USD 1 as USD 1/β and a delayed USD 1 at face value. Since 1/β > 1, it will always be the case that a TI player will perceive the immediate USD 1 as providing higher utility than the delayed USD 1. The player creating the crosscommitment device must therefore pay 1/β more when she targets the TI player’s delayed payoffs.
v_{B1}/β − c_{B1}/β + v_{B2} ≥ 0, or equivalently, (v_{B1} − c_{B1})/β + v_{B2} ≥ 0 = v_{B2} − c_{B1} ≥ 0.
Alternatively, the crosscommitment can target player B’s immediate bargaining costs. Player A will have some control over the timing in which player B incurs bargaining costs. To the extent that she can, she will have an incentive to shift some or all those bargaining costs to period 2, such that: (v_{B2} − c_{B2}) ≥ c_{B1}/β. Player A can also take steps to reduce overall bargaining costs.
Suppose that to consummate a bargain, player B must review and sign a contract. Player A can choose to deliver a complex contract that might require B to incur immediate bargaining costs of c_{B1} to comprehend it. Alternatively, A can deliver an easier to comprehend standard form contract with a lower c_{B1}. Player A can also make an interestfree loan to player B to pay for some or all his immediate bargaining costs. This is just another way to transform immediate costs into delayed ones. This sort of crosscommitment device is much more common than it may first appear. For example, trade credit is commonly used in commercial transactions—one reason is that it reduces the immediate costs of purchasing goods. Additionally, retailers pay credit card processing fees so they can offer consumers immediate utility from a purchase while avoiding the disutility from having to pay immediately in cash.
4.4. CrossCommitment Rents
As we will now see, having control over the relative timing of payoffs allows not only for benevolent crosscommitment devices, but also for nonbenevolent rentextractions. Assume that a counterpartysophisticated player A and a naïve TI player B are dividing π = 1000. Without loss of generality, we continue to assume that δ = 1. Player B has an immediacy multiplier 1/β = 2. Let player B have immediate bargaining costs, c_{B1} = 50, and an outside option with a delayed, period2 reward of 75. In period 0, the naïve player B makes a longterm decision to reject any proposal unless he gets bargaining rewards v_{B} ≥ 125, so that he receives a net bargaining surplus of at least 125 − 50 = 75 (the value of his outside option).
Suppose that in period 1, a nonbenevolent player A offers player B an immediate bargaining reward, v_{B1} = 87.50. Player B will accept the bargain, which he perceives as (2 × 87.50) − (2 × 50) = 175 − 100 = 75. In period 0, the naïve player B believed incorrectly that in period 1 he might have a β = 1 and might thus reject an offer of 87.50 − 50 = 37.50 < 75. Player A uses her counterpartysophistication to extract a sophisticationrent of 37.50 from the naïve player B. This is an example of overconsumption by player B. From a longterm perspective, he had decided not to accept such a bargain, but when faced with the prospect of immediate gratification he gave in.
4.5. Sophisticated TI Players and PrePlay Commitment
Recall that sophisticated TI players are more likely to suffer large welfare losses in instances involving immediate bargaining rewards, such as the ones in the previous subsection. Given the potential that a nonbenevolent counterparty will try to extract crosscommitment rents, one might expect that sophisticated TI players will take prophylactic actions—such as engaging in preplay commitment. More generally, if one allows for a first round of play before the bargaining game proper, two sophisticated TI players will have an incentive to engage in preplay commitment (alone, with third parties, or with their counterparty). It follows that these sophisticated TI players enter the bargaining game proper with TC preferences. They can thus be modeled using standard bargaining models.
4.6. Doubled Commitment: Targeting SelfControl and Bargaining Power
Commitment devices that commit a player to a path of play can increase the player’s bargaining power. In certain cases, bargaining and selfcontrol commitment devices may act as complements, but they may also create externalities. For example, a commitment device that forces a player to keep to her longterm TC plan can also indirectly increase her bargaining power. Additionally, one player’s bargaining commitment can act as a crosscommitment for the other player. On the other hand, a bargaining commitment device may exacerbate selfcontrol problems. For example, a bargaining commitment that might require a TI player to incur an immediate cost may exacerbate the player’s incentive to procrastinate. Thus, all other things being equal, bargaining commitments of TI players are less credible than those of their TC counterparts.
5. Extending the TI Bargaining Model
5.1. Rubinstein Bargaining with TimeInconsistent Players
So far, we have focused on relatively straightforward bargaining scenarios and seen that incorporating TI players into bargaining models designed for TC players is neither an easy nor a straightforward task. Incorporating TI players into the more complex Rubinstein bargaining model creates additional complications. Given that the growing literature on TI bargaining has given great attention to the Rubinstein model, this section extends some of the arguments developed above to the context of sequential bargaining. The goal is not to develop a fullfledged model, but to identify a set of key factors that any such model needs to address.
Let the proposer and responder have TI preferences with identical immediacy multipliers 1/β. The players make a longterm decision in period 0 about their planned course of action in periods 1 through n. From the perspective of period 0, all bargaining payoffs are in the future and thus both players make longterm plans to agree to a bargain in period 1. In period 1, however, the TI proposer and responder give added weight to any immediate costs and immediate rewards and will have an incentive to override their longterm plans if doing so might yield an immediacy premium that is greater than the delayed loss. We first analyze two contexts in which TI preferences might play no role and in which the standard bargaining model might apply. In the first context, the players can agree to a bargain in period 1, at zero cost, and all bargaining costs and bargaining rewards are received in period 2. We can apply Proposition 1 and conclude that the two TI players might reach the same bargaining result as two TC players. In the second context, both players are sophisticated (or partially naïve, but sufficiently sophisticated) about their future immediacy multiplier and engage in preplay commitment in period 0 to assure TC behavior. One possibility is for the players to agree to a contract about how they might bargain in period 1. However, players may choose to engage in preplay commitment with a third party instead, particularly if they are concerned that the other player is counterpartysophisticated and may try to extract crosscommitment rents.
Suppose now that the players are both naïve about their own immediacy multiplier and counterparty naïve about the other player’s multiplier. Suppose first that to enter a bargain the proposer must incur immediate bargaining costs in period 1 in order to produce delayed bargaining rewards in the following period. In period 1, the TI proposer overrides her longterm plan. She makes a proposal that will give her a greater portion of the delayed rewards to make up for the added weight that she now gives to the immediate bargaining costs. The TI responder might reject this bargain. From the perspective of period 1 looking at period 2, the TI responder might be making the same longterm decision as she did in period 0 (albeit visàvis a smaller pie). She thus believes that in period 2 she might make a proposal that might give her the standard share of the surplus predicted by the Rubinstein model and that the other player might accept. However, in period 2, she faces the same immediate bargaining costs and delayed rewards. She thus makes an offer that gives her a larger share of the surplus (for the same reasons as the initial proposer in period 1), and again the responder rejects. Assuming no learning, the players repeat the same set of offers period after period and never agree to a bargain.
Suppose instead that if the players reach a bargain in period 1, they get immediate bargaining rewards and incur delayed costs in period 2. For example, in period 2, the players must pay lawyers to draft a contract to memorialize the period1 bargain. Again, both players are naïve about themselves and counterparty naïve about each other. Now both players reach a bargain in period 1. In period 1, both players give added weight to the immediate bargaining rewards and not only might accept the initial proposed bargain, but each might be willing to accept less. The same result might hold if both players were sophisticated about their own immediacy multiplier. Even though this is a case in which sophisticated players face immediate rewards, the players correctly predict that in period 1 they will have an enhanced incentive to follow through with their longterm decisions.
These results underline the importance of drawing a clear distinction between immediatecosts and immediaterewards bargaining scenarios. TI bargaining models that fail to fully account for the relative timing of payoffs predict that players might reach the same bargain in both immediatecosts and immediaterewards scenarios. However, bargaining delays and bargaining breakdowns do not arise when bargaining rewards are received immediately and bargaining costs are delayed. Moreover, even in contexts in which all payoffs are received immediately, if the net surplus is sufficiently high or the immediacy multipliers are not too great, TI players might reach a bargain immediately.
5.2. Exogenous CrossCommitments
TI bargaining delays and breakdowns can produce welfare losses for the bargainers and third parties. Both types of losses are important from a social welfare maximizing perspective. For example, a TI bargaining breakdown that results in a strike can produce welfare losses for the employer, employees, customers, suppliers, and other third parties. Third parties who face large losses from bargaining delays and breakdowns will have an incentive to adopt crosscommitment devices to cause TI players to internalize the negative externalities.
5.3. Repeated, TimeInconsistent Bargaining Delays
5.4. Entry and Exit into Bargaining Relationships
5.5. Time Interval between Bargaining Periods
TI bargainers are more likely to engage in TI behavior the shorter the delay between offers. All other things being equal, shorter delays will lead to lower delayed losses from procrastination and overconsumption. Suppose that two TI players are procrastinating completing an onerous bargaining task. One player can choose to complete the task each day for 365 days and the other player must complete the task either today or a year from now. The latter player faces a greater delayed loss from not completing the task today—a one year delay—and thus has a greater incentive not to procrastinate. TC players will complete the task on the first day and will not procrastinate.
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