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Dissolving a partnership securely

Abstract

We characterize security strategies and payoffs for three mechanisms for dissolving partnerships: the Texas Shoot-Out, the \(K+1\) auction, and the compensation auction. A security strategy maximizes a participant’s minimum payoff, and represents a natural starting point for analysis when a participant is either uncertain of the environment or uncertain of whether his rivals will play equilibrium. For the compensation auction, a dynamic dissolution mechanism, we introduce the notion of a perfect security strategy. Such a strategy maximizes a participant’s minimum payoff along every path of play. We show that while a player has many security strategies in the compensation auction, he has a unique perfect security strategy.

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Fig. 2

Notes

  1. 1.

    McAfee (1992) characterizes equilibrium in the Texas Shoot-Out when partners have independent private values.

  2. 2.

    Brams and Taylor (1996) and Robertson and Webb (1998) provide surveys of this literature. Chen et al. (2013) is a recent example that incorporates both equity concerns and strategic incentives into the design of cake cutting algorithms.

  3. 3.

    The bulk of Crawford’s work on divide and choose schemes is concerned with Nash rather than maximin play, e.g., Crawford (1977, 1979, 1980b).

  4. 4.

    These mechanisms also efficiently dissolve partnerships when all the players follow security strategy.

  5. 5.

    See, for instance, Raiffa (1982) p. 297 which develops a numerical example.

  6. 6.

    In an independent private values setting, the equilibrium in strictly increasing strategies of the \(K+1\) auction where \(K=0\) is also an equilibrium of the Tie-Favoring Auction.

  7. 7.

    In the event that \(b_{\mathrm{f}}=b_{\mathrm{s}}\), the winner is selected randomly from among the high bids.

  8. 8.

    This is the difference between bidder i’s demand and the compensation \( p_{1}\) given to the rival who dropped.

  9. 9.

    Player 2’s equilibrium strategy coincides with his security strategy.

  10. 10.

    See Cramton et al. (1987).

  11. 11.

    The auction is not ex-post efficient since \(\beta (x)\ne \bar{\beta }(x)\). The probability that the allocation is inefficient is 1 / 6.

  12. 12.

    See Van Essen and Wooders (2016).

  13. 13.

    If at round \(k^{\prime }\) bidder i bids \(p_{k^{\prime }}\), then he ties with another bid. We break the tie in the other bidder’s favor so that i does not drop out. This is justified since we consider i’s worst-case payoff.

References

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Correspondence to John Wooders.

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John Wooders is grateful for financial support from the Australian Research Council’s Discovery Projects funding scheme (Project Number DP140103566).

5 Appendix

5 Appendix

Proof of Proposition 0

The proof is well known and is only included for completeness. It is easy to verify that for the given strategies the players each obtain a payoff of at least \(x_{1}/2\) and \( x_{2}/2\), respectively.

We show there is no strategy which guarantees player 1 more than \(x_{1}/2\). Consider a strategy \(\beta ^{1}\) such that \(\beta ^{1}(x_{1})>x_{1}/2\) for some \(x_{1}\). If player 1 with value \(x_{1}\) sets a price \(\beta ^{1}(x_{1})=p\) and player 2 chooses Sell, then player 1’s payoff is \( x_{1}-p<x_{1}/2\). Likewise, if \(\beta ^{1}(x_{1})=p<x_{1}/2\) for some \(x_{1}\), then player 1 obtains a payoff less than \(x_{1}/2\) if \(\beta ^{2}(x_{2};p)=\) “Buy”.

Likewise, if \(\beta ^{2}(x_{2},p)=\)“Buy” for some \(p>x_{2}/2\), then player 2 obtains a payoff \(x_{2}-p<x_{2}/2\) if player 1 offers price p. \(\square \)

Proof of Proposition 1

For symmetric mechanisms, we can write \(v_{i}(\hat{x}_{i},x_{-i},\hat{\beta }^{i},\beta ^{-i})\) as \(v(\hat{x} _{i},x_{-i},\hat{\beta }^{i},\beta ^{-i})\).

Suppose to the contrary that bidder i with value \(\hat{x}_{i}\) can guarantee more than \(\hat{x}_{i}/N\), i.e., he has a (possibly mixed) strategy \(\hat{\beta }^{i}\) such that

$$\begin{aligned} v(\hat{x}_{i},x_{-i},\hat{\beta }^{i},\beta ^{-i})\ge \bar{v}(\hat{x}_{i})> \frac{\hat{x}_{i}}{N}\quad \forall x_{-i},\beta ^{-i}. \end{aligned}$$

Since the inequality holds for all \(x_{-i}\) and \(\beta ^{-i}\), then it holds in particular for \(\hat{x}_{-i}=(\hat{x}_{i},\ldots ,\hat{x}_{i})\) and \(\hat{ \beta }^{-i}=(\hat{\beta }^{i},\ldots ,\hat{\beta }^{i})\), i.e.,

$$\begin{aligned} v(\hat{x}_{i},\hat{x}_{-i},\hat{\beta }^{i},\hat{\beta }^{-i})\ge \bar{v}( \hat{x}_{i})>\frac{\hat{x}_{i}}{N}. \end{aligned}$$

Summing across the N bidders, the total payoff when all bidders have a value of \(\hat{x}_{i}\) and follow the same strategy \(\hat{\beta }^{i}\) is \(Nv( \hat{x}_{i},\hat{x}_{-i},\hat{\beta }^{i},\hat{\beta }^{-i})\), which is greater than \(\hat{x}_{i}\). This contradicts that the mechanism is payoff feasible. \(\square \)

Proof of Proposition 2

We need to establish two facts: (i) \(\bar{\beta }^{i}(x_{i})=x_{i}\) guarantees bidder i a payoff of at least \(x_{i}/N\), and (ii) \(\bar{\beta }^{i}(x_{i})=x_{i}\) is the unique security strategy.

Part (i): Suppose that \(\bar{\beta }^{i}(x_{i})=x_{i}\). If bidder i wins, then he obtains a payoff of

$$\begin{aligned} x_{i}-\frac{N-1}{N}\left[ Kb_{\mathrm{s}}+(1-K)x_{i}\right] \ge x_{i}/N, \end{aligned}$$

where the inequality holds since \(x_{i}\ge b_{\mathrm{s}}\) as \(x_{i}\) is the winning bid. If bidder i loses, then he obtains

$$\begin{aligned} \frac{1}{N}\left[ Kb_{\mathrm{s}}+(1-K)b_{\mathrm{f}}\right] \ge x_{i}/N, \end{aligned}$$

where the inequality holds since \(b_{\mathrm{f}}\ge b_{\mathrm{s}}\ge x_{i}\). His payoff, therefore, is at least \(x_{i}/N\). By Proposition 00, \(\bar{\beta } ^{i}(x_{i})=x_{i}\) is a security strategy.

Part (ii). Suppose \(\bar{\beta }^{i}(x_{i})=x_{i}+\varDelta \) for \(\varDelta >0\). Suppose all the other bidders bid \(x_{i}+\varDelta /2\). Bidder i wins and obtains a payoff of

$$\begin{aligned} x_{i}-\frac{N-1}{N}\left[ K(x_{i}+\varDelta /2)+(1-K)(x_{i}+\varDelta )\right] <x_{i}/N, \end{aligned}$$

since \(K(x_{i}+\varDelta /2)+(1-K)(x_{i}+\varDelta )>x_{i}\) for \(K\in [0,1]\) . Suppose \(\bar{\beta }^{i}(x_{i})=x_{i}-\varDelta \) for \(\varDelta >0\). Suppose all the other bidders bid \(x_{i}-\varDelta /2\). Bidder i loses and obtains a payoff of

$$\begin{aligned} \frac{1}{N}\left[ Kb_{\mathrm{s}}+(1-K)b_{\mathrm{f}}\right] =\frac{x_{i}-\varDelta /2}{N} <x_{i}/N. \end{aligned}$$

\(\square \)

Proof of Proposition 4

S \(\bar{\beta }^{i}\) that satisfies the conditions of the Proposition. Let \(x_{-i}\) and \(\beta ^{-i}\) be arbitrary.

We first show that the bids of bidder i are in the interval \([\frac{x_{i}}{ N}+p_{k-1},\frac{kx_{i}}{N}]\) for rounds k where he remains active. In round 1, \(p_{0}=0\) and \(\bar{\beta }_{1}^{i}(x_{i};\mathbf {p}_{0})=x_{i}/N\) is determined. Suppose i does not drop at round \(k-1\), i.e., \(p_{k-1}\le \bar{\beta }_{k-1}^{i}(x_{i};\mathbf {p}_{k-2})\). Then, \(\bar{\beta } _{k-1}^{i}(x_{i};\mathbf {p}_{k-2})\le (k-1)x_{i}/N\) implies

$$\begin{aligned} \frac{x_{i}}{N}+p_{k-1}\le \frac{x_{i}}{N}+\frac{(k-1)x_{i}}{N}=\frac{kx_{i} }{N}, \end{aligned}$$

and thus the interval \([\frac{x_{i}}{N}+p_{k-1},\frac{kx_{i}}{N}]\) is non-empty and \(\bar{\beta }_{k}^{i}(x_{i};\mathbf {p}_{k-1})\in [\frac{ x_{i}}{N}+p_{k-1},\frac{kx_{i}}{N}]\) at round k.

Next, we show that bidder i obtains a payoff of at least \(x_{i}/N\) with \( \bar{\beta }^{i}\). Suppose bidder i drops at some round k. Then, \(\bar{ \beta }_{k}^{i}(x_{i};\mathbf {p}_{k-1})\ge x_{i}/N+p_{k-1}\) implies bidder i’s payoff is

$$\begin{aligned} \bar{\beta }_{k}^{i}(x_{i};\mathbf {p}_{k-1})-p_{k-1}\ge \frac{x_{i}}{N}. \end{aligned}$$

If bidder i wins the auction, then \(p_{N-1}\le \bar{\beta }_{N-1}^{i}(x_{i}; \mathbf {p}_{N-2})\le \frac{(N-1)x_{i}}{N}\) and hence bidder i’s payoff is

$$\begin{aligned} x_{i}-p_{N-1}\ge x_{i}-\frac{(N-1)x_{i}}{N}=\frac{x_{i}}{N}. \end{aligned}$$

In either case, bidder i’s payoff is at least \(x_{i}/N\). By Proposition 0, a bidder can guarantee at most \(x_{i}/N\), and thus \(\bar{\beta }^{i}\) is a security strategy.

We now establish that every security strategy satisfies the conditions of the proposition. Suppose to the contrary that \(\bar{\beta }^{i}\) is a security strategy and \(\bar{\beta }_{k}^{i}(x_{i};\mathbf {p} _{k-1})<x_{i}/N+p_{k-1}\) for some k, \(x_{i}\), and \(\mathbf {p}_{k-1}\) such that \(p_{m}\le \)\(\bar{\beta }_{m}^{i}(x_{i};\mathbf {p}_{m-1})\) for \( m=1,\ldots ,k-1\). We show that there are bids for the other bidders such that player i’s payoff is less than \(x_{i}/N\). Consider bids for the other bidders such that (i)\(\ \mathbf {p}_{k-1}\) is the sequence of dropout prices, and (ii) the bids at round k exceed \(\bar{\beta }_{k}^{i}(x_{i};\mathbf {p} _{k-1})\). Then, bidder i drops at round k and his payoff is \(\bar{\beta } _{k}^{i}(x_{i};\mathbf {p}_{k-1})-p_{k-1}<x_{i}/N\), which contradicts that \( \bar{\beta }^{i}\) is a security strategy.

Suppose that \(\bar{\beta }^{i}\) is a security strategy and \(\bar{\beta } _{k}^{i}(x_{i};\mathbf {p}_{k-1})>kx_{i}/N\) for some k, \(x_{i}\), and \( \mathbf {p}_{k-1}\) such that \(p_{m}\le \)\(\bar{\beta }_{m}^{i}(x_{i};\mathbf {p }_{m-1})\) for \(m=1,\ldots ,k-1\). Consider bids of the other bidders such that (i)\(\ \mathbf {p}_{k-1}\) is the sequence of dropout prices, (ii) some bidder \(j\ne i\) drops at round k at price \(p_{k}=kx_{i}/N+\varDelta \) for \( \varDelta >0\) and \(p_{k}<\bar{\beta }_{k}^{i}(x_{i};\mathbf {p}_{k-1})\), and (iii) for rounds \(k^{\prime }>k\) the remaining bidders bid \(\frac{x_{i}}{N} +p_{k^{\prime }-1}\).

There are two cases. First, if bidder i drops at some round \(k^{\prime }>k\) at a price strictly less than \(\frac{x_{i}}{N}+p_{k^{\prime }-1}\), then his payoff is \(\bar{\beta }_{k^{\prime }}^{i}(x_{i};\mathbf {p}_{k^{\prime }-1})-p_{k^{\prime }-1}<x_{i}/N\), which contradicts that \(\bar{\beta }^{i}\) is a security strategy.Footnote 13 Second, if bidder i wins the auction, then he pays

$$\begin{aligned} p_{N-1}=p_{k}+(p_{N-1}-p_{k})=\frac{kx_{i}}{N}+\varDelta +(N-k-1)\frac{x_{i}}{N} >\frac{(N-1)x_{i}}{N}, \end{aligned}$$

and hence his payoff is \(x_{i}-p_{N-1}<x_{i}/N\), which contradicts that \( \bar{\beta }^{i}\) is a security strategy. \(\square \)

Proof of Proposition 5

Using the same argument as in Proposition 1, it is immediate that each bidder i’s security payoff is at most \((x_{i}-p_{0})/n\). We show the given strategy achieves \((x_{i}-p_{0})/n\) , and therefore the given strategy is a security strategy in \(\varGamma (n,p_{0})\) and the security payoff is \((x_{i}-p_{0})/n\).

Let \(x_{-i}\) and \(\beta ^{-i}\) be arbitrary, and let \(p_{1},\ldots ,p_{n-1}\) be the sequence of dropout prices that results. Suppose that bidder i is not among the first \(\hat{k}-1\) bidders to drop. We show, by induction, that for \(k\in \{1,\ldots ,\hat{k}-1\}\) we have (i) \(p_{k}-p_{0}\le k(x_{i}-p_{0})/n\) and (ii) \(p_{k}-p_{k-1}\le (x_{i}-p_{k-1})/(n-k+1)\). Assume \(x_{i}>p_{0}\). Since bidder i is not the first to drop, then

$$\begin{aligned} \bar{\beta }_{1}^{i}(x_{i};p_{0})=\frac{x_{i}-p_{0}}{n-1+1}+p_{0}\ge p_{1}, \end{aligned}$$

i.e.,

$$\begin{aligned} p_{1}-p_{0}\le \frac{x_{i}-p_{0}}{n}. \end{aligned}$$

Hence, (i) and (ii) hold for \(k=1\).

Assume that (i) and (ii) hold for some \(k^{\prime }<\hat{k}-1\). We show they hold for \(k^{\prime }+1\). By the induction hypothesis, \(p_{k^{\prime }}-p_{0}\le k^{\prime }(x_{i}-p_{0})/n\) and hence \(k^{\prime }<n\) and \( x_{i}>p_{0}\) imply \(p_{k^{\prime }}-p_{0}\le x_{i}-p_{0}\), i.e., \( p_{k^{\prime }}\le x_{i}\). Since bidder i did not drop at \(k^{\prime }+1\le \hat{k}-1\), then

$$\begin{aligned} \bar{\beta }_{k^{\prime }+1}^{i}(x_{i};\mathbf {p}_{k^{\prime }})=\frac{ x_{i}-p_{k^{\prime }}}{n-(k^{\prime }+1)+1}+p_{k^{\prime }}\ge p_{k^{\prime }+1}, \end{aligned}$$

which establishes (ii) for \(k=k^{\prime }+1\). This inequality can be rewritten as

$$\begin{aligned} p_{k^{\prime }+1}-p_{0}\le \frac{x_{i}+(n-k^{\prime }-1)p_{k^{\prime }}}{ n-k^{\prime }}-p_{0}. \end{aligned}$$

By the induction hypothesis, \(p_{k^{\prime }}\le k^{\prime }(x_{i}-p_{0})/n+p_{0}\) and hence

$$\begin{aligned} p_{k^{\prime }+1}-p_{0}\le \frac{x_{i}+(n-k^{\prime }-1)(\frac{k^{\prime }(x_{i}-p_{0})}{n}+p_{0})}{n-k^{\prime }}-p_{0}=\frac{k^{\prime }+1}{n} (x_{i}-p_{0}), \end{aligned}$$

and (i) holds for \(k=k^{\prime }+1\).

If bidder i drops in round \(\hat{k}\), then his payoff is \((x_{i}-p_{\hat{k} -1})/(n-\hat{k}+1)\). Since \(p_{\hat{k}-1}\le (\hat{k} -1)(x_{i}-p_{0})/n+p_{0}\), then

$$\begin{aligned} \frac{x_{i}-p_{\hat{k}-1}}{n-\hat{k}+1}\ge \frac{x_{i}-\left( \frac{\hat{k}-1}{n} (x_{i}-p_{0})+p_{0}\right) }{n-\hat{k}+1}=\frac{x_{i}-p_{0}}{n}. \end{aligned}$$

If bidder i is not among the first \(n-1\) bidders to drop, then \( p_{n-1}-p_{0}\le (n-1)(x_{i}-p_{0})/n\). He wins the auction and his payoff is

$$\begin{aligned} x_{i}-p_{n-1}\ge x_{i}-\left( \frac{n-1}{n}(x_{i}-p_{0})+p_{0}\right) =\frac{x_{i}-p_{0} }{n}\text {.} \end{aligned}$$

Hence, \(\bar{\beta }^{i}\) guarantee’s bidder i a payoff of \((x_{i}-p_{0})/n\) and is therefore a security strategy.

If \(x_{i}<p_{0}\), then bidder i’s payoff is negative if he wins the auction. We first show that \(\bar{\beta }^{i}\) guarantees bidder i a payoff of a least \((x_{i}-p_{0})/n\). Since \(\bar{\beta }_{1}^{i}\) calls for bidder i to drop immediately, his payoff is zero unless he wins the auction. The later occurs only if all \(n-1\) other bidders drop immediately and ties are broken in bidder i’s favor. In this case, bidder i’s payoff is \( x_{i}-p_{0}\). Since this occurs with at most probability 1 / n, his expected payoff is at least \((x_{i}-p_{0})/n\). \(\square \)

Proof of Proposition 6

Write \(\bar{v} _{N-(k-1),p_{k-1}}(x_{i})\) for the security payoff of a bidder with value \( x_{i}\) in the subauction \(\varGamma (N-(k-1),p_{k-1})\). Suppose that \(\bar{\beta }_{k}^{i}(x_{i};\mathbf {p}_{k-1})<(x_{i}-p_{k-1})/(N-(k-1))+p_{k-1}\) for some k, \(x_{i}\) and \(\mathbf {p}_{k-1}\) such that \(p_{0}\le p_{1}\le \cdots \le p_{k-1}\). We show that \(\bar{\beta }^{i}\) is not a perfect security strategy. In particular, we show that \(\bar{\beta }^{i}|_{\mathbf {p} _{k-1}}(x_{i})\) yields a payoff less than \(\bar{v}_{N-(k-1),p_{k-1}}(x_{i})\) for some \(x_{-i}\) and \(\beta ^{-i}\).

From Proposition 5, the security payoff of bidder i in \(\varGamma (N-(k-1),p_{k-1})\) is \(\bar{v} _{N-(k-1),p_{k-1}}(x_{i})=(x_{i}-p_{k-1})/(N-(k-1))\). Let \(x_{-i}\) and \( \beta ^{-i}\) be such that the bids of the other \(N-k\) bidders in round 1 of \( \varGamma (N-(k-1),p_{k-1})\) are greater than \(\bar{\beta }_{1}^{i}|_{\mathbf {p} _{k-1}}(x_{i})=\bar{\beta }_{k}^{i}(x_{i};\mathbf {p}_{k-1})\). Then, bidder i drops in round 1 and his payoff is

$$\begin{aligned} \bar{\beta }_{1}^{i}|_{\mathbf {p}_{k-1}}(x_{i})-p_{k-1}<\frac{x_{i}-p_{k-1}}{ N-(k-1)}+p_{k-1}-p_{k-1}=\bar{v}_{N-(k-1),p_{k-1}}(x_{i}). \end{aligned}$$

Hence, \(\bar{\beta }_{i}\) is not a perfect security strategy.

Suppose that \(\bar{\beta }_{k}^{i}(x_{i};\mathbf {p} _{k-1})>(x_{i}-p_{k-1})/(N-k+1)+p_{k-1}\) for some k, \(x_{i}\) and \(\mathbf {p }_{k-1}\) such that \(p_{0}\le p_{1}\le \cdots \le p_{k-1}\). Let \(x_{-i}\) and \(\beta ^{-i}\) be such that (i) one of the other \(N-k\) bidders in \(\varGamma (N-(k-1),p_{k-1})\) has a dropout price \(\hat{p}_{k}\) satisfying

$$\begin{aligned} \bar{\beta }_{1}^{i}|_{\mathbf {p}_{k-1}}(x_{i})>\hat{p}_{k}>\frac{ x_{i}-p_{k-1}}{N-(k-1)}+p_{k-1}, \end{aligned}$$

and (ii) the remaining bidders’ dropout prices are above \(\bar{\beta } _{1}^{i}|_{\mathbf {p}_{k-1}}(x_{i})=\bar{\beta }_{k}^{i}(x_{i};\mathbf {p} _{k-1})\). Then, bidder 1 does not drop out in round 1 of \(\varGamma (N-(k-1),p_{k-1})\), but enters the subauction \(\varGamma (N-k,\hat{p}_{k})\). From Proposition 5, the largest payoff he can guarantee himself in this subauction is \(\bar{v}_{N-k,\hat{p}_{k}}(x_{i})=(x_{i}-\hat{p}_{k})/(N-k)\). We have that

$$\begin{aligned} \frac{x_{i}-\hat{p}_{k}}{N-k}<\frac{x_{i}-\left[ \frac{x_{i}-p_{k-1}}{N-(k-1) }+p_{k-1}\right] }{N-k}=\frac{x_{i}-p_{k-1}}{N-(k-1)}<\bar{v} _{N-(k-1),p_{k-1}}(x_{i}). \end{aligned}$$

Hence, \(\bar{\beta }_{i}\) is not a perfect security strategy. \(\square \)

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Van Essen, M., Wooders, J. Dissolving a partnership securely. Econ Theory 69, 415–434 (2020). https://doi.org/10.1007/s00199-019-01177-9

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Keywords

  • Partnerships
  • Maximin
  • Auction

JEL Classification

  • D44
  • D47
  • D82