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This article was downloaded by: [University of Coruna] On: 27 October 2014, At: 01:31 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK The Journal of Mathematical Sociology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gmas20 IFF YOU WANT ME, I DON'T WANT YOU Friedel Bolle a a Europa-Universität Viadrina , Frankfurt, Germany Published online: 12 Aug 2010. To cite this article: Friedel Bolle (2004) IFF YOU WANT ME, I DON'T WANT YOU, The Journal of Mathematical Sociology, 28:2, 57-65, DOI: 10.1080/00222500490448181 To link to this article: http://dx.doi.org/10.1080/00222500490448181 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan,

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This article was downloaded by: [University of Coruna]On: 27 October 2014, At: 01:31Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

The Journal of MathematicalSociologyPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/gmas20

IFF YOU WANT ME, I DON'TWANT YOUFriedel Bolle aa Europa-Universität Viadrina , Frankfurt, GermanyPublished online: 12 Aug 2010.

To cite this article: Friedel Bolle (2004) IFF YOU WANT ME, I DON'T WANT YOU, TheJournal of Mathematical Sociology, 28:2, 57-65, DOI: 10.1080/00222500490448181

To link to this article: http://dx.doi.org/10.1080/00222500490448181

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,

sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

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IFF YOU WANT ME, I DON’T WANT YOU

Friedel Bolle

Europa-Universitat Viadrina, Frankfurt, Germany

The usual assumption in models of asymmetric information is that I know my

attributes and you do not. But sometimes the asymmetry seems to be reversed:

You (a woman, a firm) know better than I (a man, a job applicant) my market

value. This may result in a dilemma because equilibrium strategies may be

described by the title of this paper. (Iff means: if and only if.) The result

resembles the ‘‘No-Trade-Theorem’’ of Milgrom and Stokey (1982) but cannot be

derived from it.

Know yourself!

Inscription on the Temple of Apollo at Delphi

I don’t want to belong to any club that will accept me as a member.

Groucho Marx1

INTRODUCTION

‘‘You are the prettiest girl I have ever seen!’’ Should the girl addressedbelieve this compliment? And why is there such a question at all: Doesn’tshe know whether she is pretty or not?

The usual assumption in models of incomplete information is that agentsknow their own attributes but others do not2. Apparently, however, thereare exceptions: Others know better than the girl herself whether or not sheis pretty. Others know better than I whether or not I am entertaining,pleasant, and perhaps even whether I am an efficient team player in sportsas well as in the job. Critics and readers usually know better whether or not

I would like to thank two anonymous referees and in particular the editor of this journal for

very helpful suggestions to improve the paper.

Address correspondence to the author at Europa-Universitat Viadrina, Frankfurt (Oder),

Postfach 1786, 15207 Frankfurt (Oder), Germany.

1Is also credited to W. C. Fields.

2For example, Akerlof (1973); for an overview see Molho (1997).

Journal of Mathematical Sociology, 28: 57765, 2004

Copyright # Taylor & Francis Inc.

ISSN: 0022-250X print/1545-5874 online

DOI: 10.1080/00222500490448181

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the author delivered a good piece of literature or science. We may call thisphenomenon the inverse asymmetry of information3.

Does this man look like a promising husband? Is this firm well-preparedfor the rapidly changing market? Often women and other firms can betterevaluate the decisive attributes than the owners of these attributesthemselves. But the man and the firm may learn about their attractivenessfrom whether they are accepted as partners or are even wooed, or whetherthey are not. Have you yourself once realized how much your self-esteem isimproved after a nice flirt or after a paper of yours has been accepted?Pratt and Zechauser (1991, p. 7) give a related example in another field:Thus, for example, if a Massachusetts resident is asked what price he wouldaccept in exchange for the mineral rights under his home, he might say$1,000; yet if a representative of Exxon comes to his door and offers him$50,000, he is expected to refuse. ‘‘Why is a sophisticated company

offering me $50,000?’’ he wonders, and quickly concludes, ‘‘They mustknow something I don’t. My mineral rights must be worth much more thanthat.’’ The No Trade Theorem (Milgrom and Stokey, 1982) provides uswith conditions under which the mineral rights will not be sold. We willcome back to this theorem later on.

In the following, a simple model of incomplete information is investi-gated where two agents (called man and woman) offer or accept (or not) tomate. The other’s offer (or non-offer) serves as a signal about my ownattractiveness. We will see that, in this situation, a dilemma may beinvolved that is connected with the updating of beliefs. Equilibrium mayconsist of the following strategies: If I do not know my attributes I offer(accept) mating only to (by) an attractive partner; if I know that I amunattractive I offer (accept) mating also to (by) an unattractive partner.The result is:

If you want me then I must be attractive. Thus I do not want you because I see

that you are unattractive. If you hadn’t wanted me, I would have concluded

that I am unattractive and I would have accepted you—but unfortunately you

do not want me, though you would have wanted me if you only knew that you

are unattractive, too.

There is another possibility to explain the title strategy or, in particular,the Groucho Marx quote. I know that I am unbearable; so, if this club wantsme it must have members like me7with whom I do not want to socialise. Itmay be probable that Groucho Marx should be interpreted along this line,but the ‘‘inverse asymmetry’’ meaning is also possible, and this is thebroader and more interesting phenomenon.

3A nice brain teaser with such an example is Stewart (1998).

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THE MODEL AND ITS IMPLICATIONS

There are two kinds4 of men m 2 f1; 2g and two kinds of women f 2 f1; 2g.m and f are measures of individual productivities that are, however, com-pletely effective only in a partnership. m and f are drawn from a binomialdistribution with a probability of 1

2. When they meet then, under complete

information, the following game (Figure 1) is played. Complete informationmeans that M and F know the structure of the game and the individualvalues at the endpoints. Under incomplete information (Figure 2), M doesnot know the value of m; when it is his turn to decide he does not know inwhich of those nodes, which are connected by a dashed line, he is. Thesame applies for the woman when it is her turn to decide. (The di arecommon knowledge.)

First, the man M has to decide whether or not to offer the woman Fpartnership (marriage). If he decides against an offer, an outside option isrealized where both get discounted values of their own productivities.These may be interpreted as values of staying alone or as average values oflater partnerships. In the latter interpretation, and under incomplete

FIGURE 1 The Mating Game under complete information. 12< d1; d2; d3 < 1 is

assumed.

4As types in Game Theory are described by the private information which an agent has, a

man’s type is described by f and a woman’s type by m. A complete description of an agent

consists of his=her kind and type.

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information, F starts with improved information about her productivitybecause she has not received an offer.

If M decides to offer partnership to F, she may accept or reject this offer.If she accepts it, both get the average undiscounted productivities. If sherejects the offer, again an outside option is realized. If this option consistsof a future partnership, we have to keep in mind that, under incompleteinformation, both have improved their information about their productivity:F because she received an offer, M because his offer was rejected.

Under complete information, we may assume that d1 ¼ d2 ¼ d3; in thiscase di measures only the cost of the delay, i.e., di is a measure of timepreference that may be assumed to be the same for men and women. Underincomplete information, it is plausible that d1 is smaller than d2 and d3,

FIGURE 2 The Mating Game with incomplete information. The valuations at the

endpoints are as in Figure 1. Nature chooses every alternative with a probability of 14.

Under incomplete information, we assume that M knows f, but not m, and that F knows

m, but not f. In Figure 2, this is visualized; the valuations at the endpoints are left out.

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because d2 incorporates the information for the man (the woman) that he(she) has been rejected, and because d3 incorporates the information thatthe woman has been accepted for partnership. In connection with theequilibrium strategies, this information provides them with informationabout their kind which allow more profitable choices in the future. In anycase, however, we assumes d1, d2, and d3 to take values between 1

2and 1.

By the introduction of outside options, we avoid discussing a compli-cated dynamic population game in which some participants mate in the firstin the first round whereas others improve (or not) their information in thefirst round(s) and mate later.

Proposition 1: Let us assume d ¼ d1 ¼ d2 ¼ d3.

(i) The unique perfect equilibrium for 34< d < 1 is: F decides ‘‘No’’ if (m,

f)¼ (1, 2), otherwise she decides ‘‘Yes’’. M decides ‘‘No’’ if (m, f)¼ (2,1), otherwise he decides ‘‘Yes’’.

(ii) If d < 34

then M and F deciding ‘‘Yes’’ is the unique equilibrium.

Proof:

(i) F gets (mþ f)=2 if she decides ‘‘Yes’’, she gets df it she decides ‘‘No’’. If34< d < 1 then, for all combinations of m and f except for (m, f)¼ (1,

2), ‘‘Yes’’ is connected with the higher valued outcome. If M couldexpect F to say ‘‘Yes’’ then the respective rationale would apply for him.He would choose ‘‘Yes’’ except for (m, f)¼ (2, 1). For (m, f)¼ (1, 2),however, F would reject his offer; so he would be indifferent tochoosing ‘‘Yes’’ or ‘‘No’’. This indeterminacy is removed in a perfectequilibrium (Selten, 1975) where F is assumed to make ‘‘mistakes’’ withan arbitrarily small probability: Thus the weakly dominant strategy (inevery case at least as good as other strategies) ‘‘Yes’’ becomes dominant(better than other strategies).

(ii) (mþ f )=2 is always larger than df and dm. j

Note that under complete information, either M or F want partnership,never both of them reject the other one. In the case m¼ f¼ 1 both acceptpartnership under complete information, but as we will see below they maynot manage to come together under incomplete information.

Lemma 1: In a perfect equilibrium under incomplete information, ifm¼ 2 then F decides ‘‘Yes’’, if f¼ 2 then M decides ‘‘Yes’’.

Proof: The strategies described are weakly dominant. Under the ‘‘trem-bling hand’’ assumption of perfect equilibrium they are dominant. j

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For the following, it is important to note (see Figure 2) that the Game ofMating with incomplete information has no subgame. Because of Lemma 1,M and F both have two pure strategies to be considered.

MYes: Choose ‘‘Yes’’ in any case.MNo: Choose ‘‘Yes’’ if f¼ 2 and ‘‘No’’ if f¼ 1.FYes: Choose ‘‘Yes’’ in any case.FNo: Choose ‘‘Yes’’ if m¼ 2 and ‘‘No’’ if m¼ 1.

If, in equilibrium, M chooses MNo with Prob¼ 1, then F knows that f¼ 2if he offers her to mate and f¼ 1 if not. Thus it is optimal for her to acceptM if m¼ 2 and to reject his offer if m¼ 1, i.e. FNo is the best reply to MNo.

If, in equilibrium, M chooses MYes with Prob¼ 1, then F cannot concludeanything from his choice. Under FYes she earns an expectation value of12� 2þ1

2þ 1

2� 1þ1

2¼ 5

4if m¼ 1 and under Fno she earns 3

2d3. So she should

choose FNo if

d3 >5

6: ð1Þ

If, in equilibrium, F chooses FNo with Prob¼ 1 then, if f¼ 1, theexpected value of M’s return is 1

2� 2þ1

2þ 1

2� d2 under MYes and 3

2d1 under

MNo. Se he should choose MNo if

d1 >1

2þ d2

3: ð2Þ

If, in equilibrium, F chooses FYes with Prob¼ 1 then, if f¼ 1, theexpected value of M’s return is 5

8under MYes and 3

2d1 under MNo. So he

should choose MNo if

d1 >5

12ð3Þ

which is always fulfilled according to our general assumptions. Thus thereare four possible arrow (best response) diagrams:

Proposition 2: If (2) applies, then there is a unique perfect equilibriumof the Mating Game, where M and F offer (or accept) partnership only if theother is of the more productive kind.

Proof: Figure 3, Cases (a) and (b).We are not so much interested in the other cases indicated by Figure 3.

Case (c) is another unique equilibrium and Case (d) is a rather complicatedmixed strategy equilibrium where F has to update the probability of herown productivity by means of the equilibrium probability for MNo.

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Instead of that, let us have a closer look at the equilibrium of Proposition2. Such equilibrium behaviour implies that, after receiving an offer, F knowsthat she is of the more productive kind and thus she won’t accept m¼ 1. If

You want me, I don’t want you.

If she has not received an offer she knows that she is of the less pro-ductive kind and now she would be ready to accept m¼ 1. But she can’taccept without an offer, and she won’t get one because, unfortunately, theman does not know that his productivity is m¼ 1. If you don’t want me

then I would like to accept you.Under the payoffs of the mating game, efficiency—measured by the sum of

payoffs—requires that every match results in mating. Thus (MNo, FNo)where only two high productive kinds mate is the least efficient equilibrium.

COMPARISON WITH THE NO TRADE THEOREM

Milgrom and Stokey (1982)5 show in a model with pure trade that‘‘. . .regardless of the institutional structure, if the initial allocation is ex

ante Pareto-optimal (as occurs, for example, when it is the outcome of aprior round of trading on complete, competitive markets), then the receipt

FIGURE 3 Equilibrium analysis of the Mating Game under incomplete information.

5There are further papers on the No Trade Theorem (see Morris and Skiadas, 2000, and the

literature cites there); but for the comparison with the Mating Game these are not closer

candidates.

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of private information cannot create an incentive to trade’’ (p. 17). Thereason is similar to that in the example of Pratt and Zeckhauser (1991),cited in the introduction, and similar to the above ‘‘if you want me, I don’twant you.’’ Milgrom and Stokey (1982, p. 18) describe it as ‘‘. . .the merewillingness of other traders to accept their parts of the bet [a possibletrade] is evidence to at least one trader that his own part is unfavorable.’’

Though the argument is similar, the Mating Game describes a situationcompletely different from the situation of the No Trade Theorem. Thus it isnot easy to interpret it in a way so that the No Trade Theorem is applicable.What is the ‘‘initial allocation’’? That is the crucial question!

In a first attempt, we may describe the initial state by the option toremain single. Holding this option, however, i.e., not to try to trade it forpartnership, is not Pareto-optimal in the Mating Game. On the contrary: itis disadvantageous for everybody6. Without further information about theproductivity of others (the own productivity is not known anyway, it isoptimal for every man and every women to mate ‘‘blindly’’ because thenhe=she receives 3

2 on average, instead of di � 32 when staying alone.

The No Trade Theorem can start with such a ‘‘blindly mating’’ situation,i.e., men and women are engaged, or workers have signed contracts beforereceiving information about the partner. Assume that these contractscannot be resolved without both parties agreeing. In this situation the NoTrade Theorem is applicable and it tells us that no engagement and no

working contract will be resolved after partners receive additional

information. This is a rather interesting result—but this is not the story ofthe Mating Game.

One may define Pareto-optimality also from a viewpoint of completeinformation. Then it is suboptimal for two partners with the same pro-ductivity to stay alone. But, in the Mating Game with d1 > 1

2þ d 2

3, two low

productivity partners will stay alone.Let me emphasize that the result of the Mating Game cannot be derived

from the No Trade Theorem, but that the two of them supplement oneanother rather nicely with respect to the description of a class of matingproblems.

CONCLUSION

The Mating Game may be extended in several ways. Choosing generalproductivities and general probabilities of mating is simple and no real

6Milgrom and Stokey (1982) gave in a footnote the same quotation of Groucho Marx as I

did. To apply their theorem, they must assume that the state ‘‘Groucho is not a member of

a club’’ is Pareto-optimal. (I got to know about the Milgrom and Stokey paper and about their

same citation only after a first version of my paper had been written.)

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extension. On essential option is to give also women the opportunity tooffer partnership, be it in a symmetric model where both decide at thesame time or be it in an alternating offer model7. Another alternative isthe construction of a population model where men ad women mate (havechildren or separate again). If they are singles, they know theirproductivities with certain probabilities depending, firstly, on the numberof times they have (and have not) been offered partnership or are notaccepted as partners and, secondly, depending on the equilibriumstrategies.

It is difficult to prove the empirical relevance of the dilemma pointed outin this paper. One may divide this question in two subquestions: First, arethere important examples of ‘‘reversed’’ asymmetric information? Second,is there regret by, for example, some job applicants, not to have taken anearly job opportunity because—due to such an offer—they overestimatedtheir ‘‘market value’’?

The second question may be difficult to decide as well, even on the basisof interviews, because also in ‘‘normal’’ job search models there may beregret over a lost opportunity. On the other hand, there seems to be a lot ofanecdotal evidence that overconfidence, originating from early good offers,really exists.

The first question seems to have a clearer answer though, again, we haveonly anecdotal evidence. In the introduction we have outlined a lot ofexamples where ‘‘reversed’’ asymmetric information seems to be apparent.Of course, such cases are included in general models with private infor-mation, but the investigation above has shown that these situations mightdeserve more specific attention.

REFERENCES

Akerlof, G. (1970). The market for lemons: Quality uncertainty and the market mechanism.

Quartely Journal of Economics, 84, 4887500.

Malho, I. (1997). The Economics of Information—Lying and Cheating in Markets and

Organizations. Blackwell Publishers Ltd.

Milgrom, P. & Stokey, N. (1982). Informaiton, trade and common knowledge. Journal of

Economic Theory, 26, 17727.

Morris, St. & Skiadas, C. (2000). Rationalizable trade. Games and Economic Behavior, 31,

3117323.

Pratt, J. W., & Zeckhauser, R. J. (1991). Principals and Agents: An overview. In Pratt and

Zechauser (eds.) Principals and Agents: The Structure of Business, Boston, 1991.

Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in

extensive games. International Journal of Game Theory, 4, 25755.

Stewart, I. (1998). Monks, blobs and common knowledge. Scientific American, 96797.

7In such a model, one has to expect ‘‘bluffing,’’ i.e., not showing interest for an interesting

partner.

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