The Value to a Bidder of Other Would-Be Bidders Refraining from Bidding

1. Introduction

It has recently been reported [1] that an investor $(\mathrm{A}$, she) who was planning to bid for a large firm, offered to pay another would-be bidder ($\mathrm{B}$, he) $ 5M for not submitting a bid. Presumably, A believed that B values the firm more than her and thus is likely to outbid her and win the firm. $\mathrm{B}$ accepted the offer and A indeed acquired the firm. B thus preferred accepting the $ 5M to participating in the auction and, quite possibly, ending up owning the firm. Has B done the right thing? would he have accepted a lesser amount? If so, had A overpaid?

We explore these issues, assuming first that A and B are the only bidders. If there is only one bidder, a fee needs to be paid for obtaining the firm. Then we introduce another bidder, C, who participates in the auction. We assume that bidders believe that others' bids are linear functions of their valuations [2].

As well, we consider a scenario with many symmetric would-be bidders where the question is how many to attempt to pay off.

It is assumed that B's utility function is $u_B\left(x\right)=x^r, 0<r\le 1.$ In the Appendix A we use a logarithmic utility function.

In examples we assume that parties' beliefs as to others' valuations are Pareto (first kind) distributed [3].

The auctions literature widely explored the impact of number of bidders on seller's expected revenue [e.g. [4,5]]. Rather, we are concerned with the impact of the number of bidders on an individual bidder. For general discussion of auction models see Klemperer, Krishna and Filicetti et al. [6,7,8].

2. No Other Bidder

Let $X ~ F$ be $\mathrm{B}\text{'}\mathrm{s}$ subjective belief as to ${\mathrm{A}}^{\text{'}}\mathrm{s}$ valuation and $b_{\mathrm{B}}$ his own planned bid (if an auction takes place). Let $v$ be $\mathrm{B}\text{'}\mathrm{s}$ valuation of the auctioned item. Plausibly, $v>E$(A's bid). We assume that $\mathrm{B}$ believes that $\mathrm{A}\text{'}\mathrm{s}$ bid will be

$\epsilon +\theta X, X ~ F$ [2].

Thus, if $\mathrm{B}$ rejects the offer, his expected utility of wealth would be ${\left(v-b\right)}^{r}F\left[{\displaystyle \frac{b-\epsilon }{\theta }}\right],$while if he accepts, it will be the payment $\left(p\right)$.

Let ${\displaystyle \frac{b-\epsilon }{\theta }}\equiv \tilde{b}.$The lowest payment $\mathrm{B}$ would accept is the value of $p$ which satisfies

${\left(v-b^{*}\right)}^{r}F\left({\tilde{b}}^{*}\right)=p.$

Now,

$${\displaystyle \frac{d}{db}}=-r{\left(v-b\right)}^{r-1}F\left(\tilde{b}\right)+{\left(v-b\right)}^{r}f\left(\tilde{b}\right){\displaystyle \frac{1}{\theta }}=0,$$

$$\Rightarrow rF\left({\tilde{b}}_B\right)=\left(v-{\tilde{b}}_B\right)f\left({\tilde{b}}_B\right){\displaystyle \frac{1}{\theta }}.$$

$${\displaystyle \frac{db}{dr}}={\displaystyle \frac{F\left(\tilde{b}\right)}{\left(v-\tilde{b}\right)f^{\text{'}}\left(\tilde{b}\right)-\left(r+1\right)f\left(\tilde{b}\right)}}.$$

2.1. Example 1

Pareto type 1 distribution: $F\left(x\right)=1-{\left({\displaystyle \frac{k}{x}}\right)}^a, k>0, a>1, x\ge k, f\left(x\right)={\displaystyle \frac{a{k}^a}{x^{a+1}}}\downarrow x,$

$$f^{\text{'}}\left(x\right)={\displaystyle \frac{-a\left(a+1\right)k^a}{x^{a+2}}}.E\left(X\right)={\displaystyle \frac{ak}{a+1}}. \mathrm{Here} v>k.$$

So,

$$r{\tilde{b}}^{a+1}+k^a\left(a-r\right)\tilde{b}-va{k}^a=0.$$

If $a=1,$ then $\tilde{b}={\displaystyle \frac{-k(1-r)+\sqrt{k\left[{k\left(1-r\right)}^2+4rv\right]}}{ 2r}}\Rightarrow b^{*}={\displaystyle \frac{\theta \left[-k\left(1-r\right)+\sqrt{\left(\right)}\right]}{2r}}+\epsilon$.

$${\displaystyle \frac{{d\tilde{b}}^{*}}{dr}}>0 \mathrm{iff} {\tilde{b}}^{*}>{\displaystyle \frac{v\left(a+1\right)}{a+r+2}} .{\displaystyle \frac{{d\tilde{b}}^{*}}{da}}>0 \mathrm{iff} {\tilde{b}}^{*}>{\displaystyle \frac{av-1}{a-r}} .$$

$$E^{*}=\left[\sqrt{\left(\right)}-k\left(1+r\right)\right]{\displaystyle \frac{{\left[2rv+k\left(1-r\right)-\sqrt{\left(\right)}\right]}^r}{2r^2\left[-k\left(1-r\right)\right]+\sqrt{\left(\right)}}} .$$

2.2. Example 2: Power($\alpha$) on [s,c]

$$F\left(x\right)={\left({\displaystyle \frac{x-s}{c-s}}\right)}^{\alpha }, \alpha \ge 1, f\left(x\right)={\displaystyle \frac{\alpha {\left(x-s\right)}^{\alpha -1}}{{\left(c-s\right)}^{\alpha }}}$$

$$E={\displaystyle \frac{{\left(v-\theta \widehat{b}-\epsilon \right)}^r{\left(v-\theta \widehat{b}-\theta -s\right)}^{\alpha }}{{\left(c-s\right)}^{\alpha }}}.$$

$${\displaystyle \frac{dE}{d\tilde{b}}}=\theta {\left(v-\theta \widehat{b}-\epsilon \right)}^{r-1}{\left(v-\theta \widehat{b}-\epsilon -s\right)}^{\alpha -1}\bullet$$

$$r{\left[{\displaystyle \frac{\tilde{b}-s}{c-s}}\right]}^{\alpha }={\displaystyle \frac{\left(v-\tilde{b}\right)\alpha {\left(\tilde{b}-s\right)}^{\alpha -1}}{{\left(c-s\right)}^{\alpha }}}\Rightarrow \tilde{b}={\displaystyle \frac{\alpha v+rs}{a+r}}\Rightarrow b^{*}={\displaystyle \frac{\theta \left(\alpha v+rs\right)}{\alpha +r}}+\epsilon , \uparrow v, \alpha ,\downarrow r.$$

$$E^{*}={\displaystyle \frac{r^r{\alpha }^{\alpha }{\left(v-s\right)}^{r+\alpha }}{{\left(r+\alpha \right)}^{r+\alpha }{\left(c-s\right)}^{\alpha }}}, \uparrow r \mathrm{i}\mathrm{f}\mathrm{f} r>{\displaystyle \frac{\alpha }{v-s-1}}. \uparrow \alpha \mathrm{i}\mathrm{f}\mathrm{f} r<v-s-\alpha .$$

3. Should A Make an Offer?

$\mathrm{A}$ has to choose between participating in the auction (against B) or pay him and receive the object for the minimum fee, T. $\mathrm{A}\text{'}\mathrm{s}$ valuation is $w,$ and her belief as to ${\mathrm{B}}^{\text{'}}s$ valuation is captured by the distribution $G.$ A is assumed to be risk neutral. We shall assume here that $\epsilon =0$ and $\theta =1.$ Thus, if A does not attempt to exclude B

$$E_{\mathrm{A}}=\left(w-b_{\mathrm{A}}\right)G\left(b_{\mathrm{A}}\right)$$

$${\displaystyle \frac{d{E}_{\mathrm{A}}}{d{b}_{\mathrm{A}}}}=-G\left(b_{\mathrm{A}}\right)+\left(w-b_{\mathrm{A}}\right)g\left(b_{\mathrm{A}}\right)=0,$$

while if it does, $E_A=w-T-p.$

3.1. Example1 (Cont.) G ~ Pareto(k, a)

$$-\left[1-{\left({\displaystyle \frac{k}{b}}\right)}^a\right]+\left(w-b\right){\displaystyle \frac{a{k}^a}{b^{a+1}}}=0,b>k>1.$$

$${\Rightarrow b}^{a+1}+\left(a-1\right)k^{a}b-wa{k}^a=0 ,w>k>1.$$

$${\displaystyle \frac{d{b}^{*}}{da}}>0.$$

$${\displaystyle \frac{d{b}^{*}}{dk}}>0 \mathrm{i}\mathrm{f} b<{\displaystyle \frac{wa}{a-1}}.$$

Let $a=1.$ Then if B rejects offer, $b^2-wk=0 \Rightarrow b=\sqrt{wk}$ . As $w>k,$ then $b>k.$

$\Rightarrow E_A=\left(w-\sqrt{wk}\right)\left[1-{\displaystyle \frac{k}{\sqrt{wk}}}\right]=w\left(1+k^2\right)-2\sqrt{wk}.$

If B accepts offer, $E_A=w-T-p, \mathrm{s}\mathrm{o} \mathrm{s}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{d} \mathrm{n}\mathrm{o}\mathrm{t} \mathrm{m}\mathrm{a}\mathrm{k}\mathrm{e} \mathrm{o}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r} \mathrm{i}\mathrm{f}$ $k+w-2\sqrt{wk}>w-T-p,$

i.e., if $T>2\sqrt{wk}-p-w{k}^2.$

That is, if the single-bidder fee is high, then A should not attempt to exclude B. The upper bound on the fee is increasing in $w,$ and decreasing in $k.$

4. Impact of Presence of Third Bidder (C)

4.1. B Accepts Payment (and Withdraws)

$E_A=\left(w-b\right)H\left(b\right),$ where $H$ is $\mathrm{A}\text{'}$s belief-distribution as to C's bid.

$\Rightarrow {\displaystyle \frac{d{E}_A}{db}}=-H\left(b\right)+\left(w-b\right)h\left(b\right)=0.$

4.1.1. Example 1 (Cont.)

As $h\left(b\right)={\displaystyle \frac{a{k}^a}{b^{a+1}}}\downarrow b,{\displaystyle \frac{db}{dw}}={\displaystyle \frac{b}{2[b+\left(a+1\right)\left(w-b\right)]}}>0.$

$${\displaystyle \frac{d{E}_A}{db}}=-b^{a+1}+k^a\left(1-a\right)b+wa{k}^a=0.$$

When $a=1,$ then $b^{*}=\sqrt{wk} , E^{*}=w+k-\left(1+w\right)\sqrt{wk}$.

When $a=2,$ then if $b=1,$ then $w={\displaystyle \frac{k^2+1}{2k^2}}, E={\displaystyle \frac{{\left(k^2-1\right)}^2}{2k^2}}.$

If $b=2,$ then $w={\displaystyle \frac{k^2+4}{k^2}},E={\displaystyle \frac{3k^4-16k^2+16}{8k^2}}.$

$$\Rightarrow \left\{b=1\right\}\succ \left\{b=2\right\} \mathrm{i}\mathrm{f}\mathrm{f} k>\sqrt{\sqrt{7}-2 }~ 1.20.$$

When $a=3,$ then $w={\displaystyle \frac{b^4+{2k}^{3}b}{3k^3}}.$ If $b=1,$ then $w={\displaystyle \frac{{1+2k}^3}{3k^3}}, E^{*}={\displaystyle \frac{{\left(1-k^3\right)}^2}{3k^3}}.$

If $b=2,$ then $w={\displaystyle \frac{4\left(4+k^3\right)}{{3k}^3}}, E^{*}={\displaystyle \frac{{\left(k^3-8\right)}^2}{12k^3}}.$

$$b=3, w={\displaystyle \frac{27+2k^3}{k^3}}\Rightarrow E^{*}={\displaystyle \frac{{\left(k^3-27\right)}^2}{27k^3}}$$

$$\Rightarrow \left\{b=1\right\} \succ \left\{b=2\right\} \mathrm{i}\mathrm{f}\mathrm{f} k>1.50.$$

$$\left\{b=2\right\} \succ \left\{b=3\right\} \mathrm{i}\mathrm{f}\mathrm{f} k>2.50.$$

$$\left\{b=1\right\} \succ \left\{b=3\right\} \mathrm{i}\mathrm{f}\mathrm{f} k>1.96.$$

4.1.2. Example 2 (Cont.): Power($\alpha$) on [s,c]

$$H\left(x\right)={\left({\displaystyle \frac{x-s}{c-s}}\right)}^{\alpha }, h\left(x\right)={\displaystyle \frac{\alpha {\left(x-s\right)}^{\alpha -1}}{{\left(c-s\right)}^{\alpha }}}, h^{\text{'}}\left(x\right)={\displaystyle \frac{\alpha \left(\alpha -1\right){\left(x-s\right)}^{\alpha -2}}{{\left(c-s\right)}^{\alpha }}}>0.$$

$${\displaystyle \frac{d{E}_A}{db}}=-{\left({\displaystyle \frac{b-s}{c-s}}\right)}^{\alpha }+{\displaystyle \frac{\left(w-b\right)\alpha {\left(b-s\right)}^{\alpha -1}}{{\left(c-s\right)}^{\alpha }}}=0\Rightarrow {\left(b-s\right)}^{\alpha -1}\left[-\left(b-s\right)+\left(w-b\right)\alpha \right]=0$$

$$\Rightarrow b^{*}={\displaystyle \frac{w\alpha +s}{\alpha +1}} , \uparrow \alpha \Rightarrow E_A^{*}={\displaystyle \frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}{\left(c-s\right)}^{\alpha }}}, \uparrow \alpha \mathrm{i}\mathrm{f} w>{\displaystyle \frac{\left(\alpha +1\right)c-s}{\alpha }}.$$

4.2. B Does Not Accept Bribe

As to win the auction now $\mathrm{A}$ has to outbid both $\mathrm{B}$ and $\mathrm{C}$,

$$E_{\mathrm{A}}=\left(w-b\right)G\left(b\right)H\left(b\right)$$

$${\displaystyle \frac{d{E}_{\mathrm{A}}}{db}}=-G\left(b\right)H\left(b\right)+\left(w-b\right)\left[g\left(b\right)H\left(b\right)+G\left(b\right)h\left(b\right)\right]=0.$$

4.2.1. Example 1 (Cont.)

$$E=\left(w-b\right)\left[1-{\displaystyle \frac{k_G^a}{b^a}}\right]\left[1-{\displaystyle \frac{k_H^a}{b^a}}\right].$$

$${\displaystyle \frac{dE}{db}}=0\Rightarrow -b^{2a+1}+aw{\left(k_G^a+k_H^a\right)b}^a-\left(a-1\right)\left(k_G^a+k_H^a\right)b-2wa{k}_G^a{k}_H^a=0$$

If $a=1$,

$$b^3-\left(wa\widehat{k}+\tilde{k}\right)b+2w\tilde{k}=0$$

Some solutions: $\left\{b=1, w={\displaystyle \frac{\tilde{k}-1}{2\tilde{k}-a\widehat{k}}}\right\}, \left\{b=2,w={\displaystyle \frac{2\left(\tilde{k}-4\right)}{2\tilde{k}-a\widehat{k}}}\right\},$

where $k_G^2+k_H^2\equiv \overline{k}.$

If $a=2$,

$$b^5-\overline{k}{b}^3-2w\overline{k}{b}^2-{\left(\tilde{k}\right)}^2\left(4w-3\right)=0.$$

Some solutions: $\left\{b=1, w={\displaystyle \frac{3{\left(\tilde{k}\right)}^2-\overline{k}-1}{2{\left(2\tilde{k}\right)}^2-\overline{k}}}\right\}, \left\{b={\displaystyle \frac{1}{2}},w={\displaystyle \frac{1+16{\tilde{k}}^2+124\overline{k}}{256\overline{k}}}\right\},$

$$\left\{b=2,w={\displaystyle \frac{3{\left(\tilde{k}\right)}^2-4\overline{k}-16}{2\left[{\left(\tilde{k}\right)}^2-2\overline{k}\right]}}={\displaystyle \frac{3k_G^2{k}_H^2-4\left(k_G^2{+k}_H^2\right)-16}{2\left[k_G^2{k}_H^2-2\left(k_G^2{+k}_H^2\right)\right]}}\right\}.$$

4.2.2. Example 2 (Cont.)

$$G\left(b\right)={\left({\displaystyle \frac{b-s}{c-s}}\right)}^{{\alpha }_1}, H\left(b\right)={\left({\displaystyle \frac{b-s}{c-s}}\right)}^{{\alpha }_2}, w>s.$$

$$\Rightarrow E=\left(w-b\right){\left({\displaystyle \frac{b-s}{c-s}}\right)}^{{\alpha }_1+{\alpha }_2}$$

$${\displaystyle \frac{dE}{db}}=0\Rightarrow -\left(b-s\right)+\left({\alpha }_1+{\alpha }_2\right)\left(w-b\right)=0$$

$$\Rightarrow b^{*}={\displaystyle \frac{\left({\alpha }_1+{\alpha }_2\right)w+s}{{\alpha }_1+{\alpha }_2+1}} ,\uparrow w, s, \left({\alpha }_1+{\alpha }_2\right),$$

independent of $c$.

${\Rightarrow E}^{\mathrm{*}}={\displaystyle \frac{{\left({\alpha }_1+{\alpha }_2\right)}^{{\alpha }_1+{\alpha }_2}{\left(w-s\right)}^{{\alpha }_1+{\alpha }_2+1}}{{\left({\alpha }_1+{\alpha }_2+1\right)}^{{\alpha }_1+{\alpha }_2+1}{\left(c-s\right)}^{{\alpha }_1+{\alpha }_2}}},\uparrow w,\downarrow c,\uparrow s$ iff $c-s<{\alpha }_1+{\alpha }_2$ [which obviously holds if

$$\left(s,c\right)=\left(\mathrm{0,1}\right)].$$

5. How Many Symmetric Would-Be Bidders Should One Attempt to Convince to Refrain from Bidding?

Suppose that A is aware of n other serious symmetric would-be bidders. Suppose she try to pay off $k$ of them, $k=0, 1\dots ,n$. We shall assume that $F\left(x\right)=x^2, 0\le x\le 1 \left[Power\left(2\right)\right],$ and that $u\left(x\right)=x^r, 0<r\le 1.$Assume that those paid off indeed refrain from bidding.

A's objective (expected utility) is then ${\left(-kp+w-b\right)}^r b^{2\left(n-k\right)}\Rightarrow b_k^{*}={\displaystyle \frac{2\left(w-kp\right)\left(n-k\right)}{2\left(n-k\right)+r}} .$Substituting $b_k^{*}$, objective becomes $r^r{\left(w-kp\right)}^{2\left(n-k\right)+r}{\left(n-k\right)}^{2\left(n-k\right)}{2}^{2\left(n-k\right)}.$ It can be shown that $k^{*}$ equals ${\displaystyle \frac{w}{p}}$ [note that $\mathrm{w}\gg \mathrm{p}].$

6. Concluding Remarks

A bidder's preference to bid against as few others as possible is clear. The question is how much is a reduction of one bidder worth to her, and how much should the other bidder demand for refraining from bidding. We explored these issues in situations when there is only one other potential bidder and when there are two, one of whom is the potential target for bribe. We exemplify it for Pareto and power valuation distributions. We assumed that the target bidder is risk averse with power or logarithmic utility functions. It turned out that in the motivating case it is quite possible that the bidder paid too much, as the targeted would-be bidder would have agreed to withdraw for a lesser amount. If such practice turns out to be legal in the relevant jurisdiction, then in auctions with few potential bidders the efficacy of paying the most "threatening" one to refrain from bidding should be examined. That, as we have seen, required deriving the optimal bid in case the targeted bidder declines the offer. One can expect that at times both would-be bidders will attempt to pay each other to withdraw, and the consequences of such a game need to be explored.

We also addressed the determination of how many would-be bidders to attempt to exclude in a symmetric scenario.

Appendix A.1. Example 1 (Cont.)

Pareto: $F\left(x\right)=1-{\left({\displaystyle \frac{k}{x}}\right)}^a,k>0,a>1,x\ge k$.

$$E_B=\left[log\left(v-b\right)\right]\left[1-{\left({\displaystyle \frac{k}{b}}\right)}^a\right]$$

$${\displaystyle \frac{d{E}_B}{db}}=-{\displaystyle \frac{1}{v-b}}\left[1-{\left({\displaystyle \frac{k}{b}}\right)}^a\right]+{\displaystyle \frac{a{k}^{a}log\left(v-b\right)}{b^{a+1}}}=0$$

$$\Rightarrow a{k}^a\left(v-b\right)log\left(v-b\right)-b^{a+1}+k^{a}b=0$$

$${\displaystyle \frac{d{b}^{\mathrm{*}}}{dk}}>0. {\displaystyle \frac{d{b}^{\mathrm{*}}}{da}}>0 \mathrm{if} 1<b<v-1 \mathrm{and} k>1.$$

If $a=1,$ then $k\left[\left(v-b\right)log\left(v-b\right)+b\right]=b^2.$

Thus, if $b=v-2\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}k={\displaystyle \frac{{\left(v-2\right)}^2}{v}}.$

For $a=2,$ if $b=v-2,$ then $={\displaystyle \frac{{\left(v-2\right)}^{\frac{3}{2}}}{{\left(6-v\right)}^{\frac{1}{2}}}},2<v<6.$

Appendix A.2. Example 2 (Cont.) Power(α) on [s, c]

$$E_B=\left[log\left(v-b\right)\right]{\left[{\displaystyle \frac{b-s}{c-s}}\right]}^{\alpha }.$$

$${\displaystyle \frac{d}{db}}=0\Rightarrow {\displaystyle \frac{-\left(b-s\right)}{v-b}}+\alpha log\left(v-b\right)=0.$$

If $b=v-2,$ then $\alpha ={\displaystyle \frac{v-s-2}{2}},2<v-s<4.$