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The hold-out problem often occurs in debt-restructuring or in urban development.

The hold-out problem is defined where an agent, for example a land developer, must negotiate with many lot owners and each lot owner must provide their consent in order for a project to proceed. However, since the agents know that their consent is required they can extract a portion of the producer surplus in excess of their opportunity costs. The net effect is that such large-scale projects are underproduced.

There are legal remedies to address this underproduction. Eminent domain is a legal tool to overcome hold-out problems, however, it does not consider pareto-efficiency. It's not at all clear whether developers that exercise eminent domain are increasing social welfare or simply have better influence over a political process to transfer resources from one group to another.

Ideally, we would like a voluntary bargaining solution that all (or most) parties agree to that produces a socially efficient amount. A bargaining solution should always be arrived at when the social utility of the production exceeds the opportunity costs of each individual and the producer cost.

What bargaining "game" could be designed (and enforced or enabled by a legal process) so that such a procedure could be applied to hold-out problems?

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Very good question. I think there's something in some of the more esoteric auction models that may be useful. I'll have a hunt... –  Turukawa Nov 8 '11 at 6:42
@Turkuawa - that would make sense... To your point, there are various auctions designed for bidders to bid their true reservation price –  Quant Guy Nov 8 '11 at 12:37
I can give you an answer, but I'm afraid it'll be a little speculative. There is a good body of research, but it simply repeats the quandaries you mention in your question (Eminent domain, with all the moral hazard involved). The consequences are also interesting... –  Turukawa Nov 8 '11 at 17:18
I've updated my answer to include a recent paper by Miceli published in Public Choice. He models a noncooperative strategic game with an eye toward applicability for housing development. You might want to check it out. –  jrhorn424 Nov 15 '11 at 22:26
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2 Answers

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The hold-out problem is much studied and has even been associated with IP disputes (especially relevant in an age where a complex device like a mobile phone can involve license agreements between hundreds of companies).

Some of the consequences being identified as a result of the hold-out problem include urban sprawl (easier to develop green-fields sites), urban decay (hold-outs prevent urban renewal), and even disruption of the innovation and investment cycle (hold-outs over-value their IP contribution and innovators then refuse to invest).

In Imminently Eminent: A Game Theoretic Analysis of Takings Since Kelo v. City of New London, Alex Hornaday proposes the following game:

  • Game participants include a Developer and an Owner;
  • Developer values Owner's property at $x$;
  • Owner values property in a range $p_1$ to $p_2$ such that $p_1 < x < p_2$;
  • Basic play:
    1. Developer offers price $x_1$;
    2. Owner either accepts or rejects $x_1$;
    3. if Owner rejects $x_1$, Developer may quit bargaining or offer price $x_2$;
    4. if Developer offers price $x_2$, Owner either accepts or rejects $p_2$;
    5. if Owner rejects $x_2$ and the Developer has the power of eminent domain, Developer may then choose either to do nothing or to condemn;

Hornaday also includes a penalty which the Owner would incur during round 2. He acknowledges that much of this is Pareto-inefficient as either (or both) parties are forced into a loss following an unsatisfactory conclusion.

The oversimplification here is that hold-outs tend to occur when multiple parties need to negotiate. In that case the last part to negotiate a deal is able to exert a monopoly pricing power over the other parties.

Eminent domain certainly offers the eventuality of a result (from the Developer's perspective) but can be distortionary (since either party may have insufficient negotiating power to secure an ideal result).

My answer, at this point, becomes speculative and I offer this untested as a potential model for achieving a "better" response amongst multiple parties. It is based on the concept of optimal reservation prices in auctions.

Supplementing Hornaday's game as follows:

  1. Developer makes a sealed reserve offer, $r_1$, for all properties from all Owners;
  2. All Owners make a sealed pooled reserve request, $r_2$, for all properties;
  3. Negotiator establishes whether $r_1$ and $r_2$ are aligned;
  4. If not, then an assessment (legal and social) must take place (at Developer's expense) if the Developer wishes to continue to assess whether eminent domain is even possible (not a decision, however, so pertaining to a negotiation only);
  5. If negotiation is possible, or if eminent domain is a viable solution to negotiation, then price negotiation can continue with all parties knowing where they stand.

We can now continue with the original game. I believe that there need to be sufficient inducements and penalties for the Owners (inducements for early settlement, penalties for holding out) to balance the Developer's costs in the process.

Obviously, if the reserve prices make a settlement unlikely and eminent domain would not be possible, then the Developer must walk away and pocket the costs of the process to that point.

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Nice analysis. You would need some mechanism in step #2 so that the owners optimal bid is their reservation price (and is robust to potential collusion). Obviously this is beyond the scope of a post like this so I appreciate your outlining a solution here! –  Quant Guy Nov 8 '11 at 20:17
The Negotiator would have to be some mutually-approved legally mandated third-party (lawyer, notary, etc.) who would act in good faith for both sides. Receive the bids / reserve prices / ... this way an audit trail is created. I figure the only way to prevent harmful collusion is to make sure everything is recorded. –  Turukawa Nov 9 '11 at 6:25
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If you're looking for a simple game, for example to teach the holdout problem to students just beginning game theory, you could do worse than start with a general prisoner's dilemma (PD) game. The 30-second explanation is that it only makes (rational) sense for one player to act if the other person is acting as well.

The holdout problem is a motivation for voluntary contribution mechanism experiments (VCMs) or "public goods" games, which is an example of a PD game in public context.

These games only speak to the private optimum of holding out (the unilateral optimum) as compared to the social optimum of cooperating (the multilateral optimum), and doesn't speak to extracting surplus. Cooperation can be ensured with contracts and other extra-game mechanisms, usually relying on the possibility of negotaition and requiring enforcement of contracts. Starting with a simple PD, including side-payments explicitly in the game structure should be sufficient for capturing the flavor of surplus extraction.

Update: Miceli has recently published a noncooperative game model for the holdout problem in Public Choice as "Free riders, holdouts, and public use: a tale of two externalities".

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