- Email: [email protected]
Bargaining rules of thumb
Journal
of Economic
Behavior
and Organization
22 (1993)
15-24.
North-Holland
Bargaining rules of thumb Robert
W. Rosenthal*
Boston University, Boston, MA 02215, USA Received
May 1992, final version
received
June 1992
A steady-state, random-matching game model is used to described bargaining in a large population, where bargainers choose rules of thumb to apply across the board in a class of bargaining situations they face. The unique population equilibrium rule distributions are computed for certain cases of the model. These are then used in an exercise to analyze the relationship between a society’s wealth and the relative frequency of haggled transactions in the society.
1. Introduction Within game theory there are two main categories of bargaining theories. Axiomatic bargaining theory emanates from Nash (1950). A typical contribution to this literature’ assumes there is a mapping on some domain of bargaining games that selects the outcome of each game in the domain; it then characterizes the mapping axiomatically. The typical axiom system prescribes both what might constitute reasonable behavior in certain of the games and what might constitute consistency of the mapping across games in the domain. Noncooperative bargaining theory is the second category. It received its main impetus from Rubinstein (1982), although precursors include Harsanyi and Selten (1972) and Stahl (1972). This literature is single-game oriented: a typical contribution posits precise rules for the bargaining process and analyzes Nash or Bayesian equilibria (or refinements thereof) of the resulting game. This paper falls outside both of the categories above and follows the general approach of Rosenthal (1993). It assumes that players face many bargaining situations over time and adopt rules of thumb to apply in all bargaining situations, rather than respond in equilibrium fashion to each single game as it occurs. Thus this theory might be seen as applying to Correspondence to: R.W. Rosenthal, Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA. *Thanks to Jonathan Eaton and Wolfgang Leininger for helpful comments. This research has been supported by National Science Foundation grant no. SES-9010246. ‘Roth (1979) contains an excellent survey of this literature through the late 1970’s. 0167-2681/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
16
R. W. Rosenthal,
Bargaining
rules of thumb
frequently-occurring, low-value bargaining situations instead of to an isolated situation in which there is a great deal at stake. It therefore shares with the axiomatic literature a concern for consistency across different games. That concern is motivated here, however, by a belief that much behavior (and in particular much bargaining behavior) is of the automatic-pilot variety and that people choose rules of thumb to apply in broad classes of situations, rather than tine tune behavior to each situation. Like the noncooperative bargaining literature, the theory has a noncooperative-equilibrium type solution concept, termed a population equilibrium. The model is of the largesociety, random-matching variety;2 so the equilibrium conditions are not Nash noncooperative equilibrium conditions, except that they are equivalent to the Nash equilibrium conditions for a particular related game. Unlike the noncooperative bargaining literature, the bargaining process is modeled here in a crude way, with no attention given to specific bargaining procedures. The model is simple enough that it is possible to describe explicitly its unique population equilibrium (at least in the versions of the model in which there are continua of choices) as a function of certain parameters. This means that it is possible to compare equilibria across societies having different characteristics. As an illustration, the model is used here to address the question of why haggling is more prevalent or intense in certain societies and in certain situations than in others. (Others, for example Riley and Zeckhauser (1983) and Wang (1991), have compared haggling with postedprice selling, but the points-of-view, models, and objectives of those studies are different from this one.) The continuum-of-actions versions of the model are spelled out in section 2, and the unique population equilbria are identified for each of these versions of the model. Section 3 contains a comparative-statics exercise, together with interpretive remarks, in which the equilibrium of a relatively poor society is compared with that of a relatively rich one. In section 4, the complications introduced by dropping the continuum-of-actions assumption are discussed. Section 5 consists of a collection of remarks about the model, the results, and interpretations. The proofs of all the results are confined to an appendix.
2. The model and its equilibria Imagine a society consisting of a large number of individuals who are randomly matched in pairs on a regular basis to play bargaining games. The games are all of the following form: Each bargainer simultaneously selects a ‘Other large-society, random-matching models of bargaining Landau (1979) and (1981), but they are considerably different details.
can be found in Rosenthal from this one in both spirit
and and
R.W. Rosenthal, Bargaining rules ofthumb
17
bargaining effort levek3 whoever’s effort level is greater wins the greater share of the object being bargained over. More precisely, in version Z” of the model each matched pair of players plays a random game k, where k indexes parameters (W,, T,J, drawn from a set K of games according to a known distribution (pJkeK. Informed of k,4 the players, call them i and j, simultaneously select effort levels ei and ej from the nonnegative real numbers R +. If e, > ej, i’s payoff is W, -e, and j’s payoff is -ej (and vice versa when i and j are reversed). In the event of a tie, both receive Tk -e,. (Of course, W, > Tk > 0, Vk E K .) Before finishing the description of the model, two aspects of this formulation require immediate emphasis. First, there are only two possible outcomes to any game: win-loss and tie; so the bargaining is indeed of an extremely crude form.’ Second, the winner pays his actual effort; so it is not exactly right to interpret an effort level as the maximum time cost the player is prepared to incur, as in the ‘war of attrition’ [e.g., Milgrom and Weber (1985)] or money cost, as in the ‘dollar auction’ [e.g., Shubik (1971), Leininger (1989)]. Here the players pay their respective effort costs independently of those chosen by their opponents. Thus the individual games resemble a sealed-bid version of the dollar auction. The idea is that players must prepare themselves beforehand for their planned efforts, and that is when the cost is incurred. (See below and section 5 for more on this.) There is assumed to be anonymity and independence in the matching process but enough information about population averages that individuals all assume their current opponent in every period is simply a new random draw from a known distribution of behaviors, the same distribution in each period. Equilibrium requires this distribution to be the true frequency distribution of the population’s actions. Finally, individuals are assumed to choose their effort levels once and for all, and not as a function of k. This is the ‘automatic-pilot’ or rule-of-thumb restriction. For justification, think of effort-level selections as attitudes or emotions that are difficult or impossible for an individual to change, as in Frank (1988). A tough bargainer is what one is or is trained to be, not what one chooses to be. In this interpretation, the force that drives the population toward equilibrium (or at least away from disequilibrium) is an evolutionary one: successful effort levels reproduce; others die out. For further justification, see Rosenthal (1993) and references therein. The effort cost should therefore be interpreted either as a per-game amortization of the one-time effort selected or the flow cost of maintaining the chosen rule. A population equilibrium of r is then a cumulative
3The word ‘effort’ should be taken as a catchall standing for different types of activities that are costly to the player but that increase the chances for success in bargaining. %Cs turns out to be irrelevant in the light of what is to come. sSection 5 revisits this assumption. Adopting 0 as the value of a loss is for expository convenience.
R.W. Rosenthal, Bargaining rules of thumb
18
distribution maximizer
function of:
ACNE
(c.d.f.) F on R +, each element
-x+WF(x)-(W-T)(F(x)-F(x-))
of whose
support
is a
VXER+,
where W -xkpk W, and T=x,p,T,. From this definition6 it should be obvious that the population equilibria of f are identical to the symmetric Nash equilibria of the symmetric twoperson one-shot game in which each player’s pure-strategy space is R, and Player l’s payoff as a function of his strategy x and 2’s strategy y is: (W-x)
if x>y,
(T-x)
Player 2’s payoff is defined Despite the discontinuity is easy to find.
if x=y,
and (-x)
by symmetry. on the diagonal,
Theorem 1. The unique population distribution on [0, W].
equilibrium
if x
the set of population
equilibria
of r is the c.d.f. of the uniform
It might be the case that exogenous constraints limit the amount of effort a player can devote to bargaining; for instance, if effort is for some reason unproductive beyond some point. We shall therefore consider the following variant of r. For the number u>O, define rU to be f with the modification that only the effort levels in [0, u] are permitted, and modify the definition of the population equilibrium in the obvious way. Theorem 2. as follows.
For each u>O, there is a unique population
(1) For u >=W, F is the c.d.f. of the uniform distribution (2) For T
equilibrium
F of T,,
on [0, W];
(xl W) for x < W(u - T)/( W - T) F(x)=
(u-T)/(W-T)
for W(u-T)/(W-T)Sx
1 forxzu (3)
For u5 T,
F(x) =
0 forx
6As c.d.f.‘s are right-continuous, assigned by F to x.
F(x)-F(x-)
simply
denotes
the probability
mass
(if any)
R. W. Rosenthal, Bargaining rules of thumb
19
Notice that in r and in the first two cases of f. the effort of each individual at equilibrium just equals his expected return from the average bargaining game. In case (3) of T,, when u < T, the upper bound on effort levels is so low that even when everyone chooses the largest possible effort level, the gains from bargaining are not exhausted on average. 3. Comparative statics The cross-societal question is: what happens to the population equilibrium when W and T either increase or decrease in size relative to the effort-level scale. To put this question in its simplest form, let us assume that T is a constant fraction TVof W (O
are based on extremely casual observation and hearsay. I have not found literature that bears on cross-society haggling frequency comparisons.
any formal
20
R. W. Rosenthal, Bargaining rules of thumb
societies a larger set of transactions is customarily haggled over (rather than sold by posted prices); (ii) the additional haggled transactions of the poorer society are at the low-value end of the spectrum; and (iii) more effort (time) is expended in haggling over the commonly haggled transactions in the poorer society. Assuming purchasing-power parity for the haggled transactions, that excess supply (and immobility) of labor should hold down the relative value of effort on average in the poorer society, and ignoring both the caveats above and (i) and (ii), the conclusion of Theorem 3 agrees with stylized fact (iii): the same object is valued relatively more highly in the poorer society, so we should expect to see more effort expended in bargaining there. On the other hand, the effects of both (i) and (ii) are to raise the average relative value of the objects haggled over in the richer society; and these effects should increase average haggling in equilibrium in the richer society, again as a consequence of Theorem 3. So, if all three stylized facts are true, our story is that the logic of Theorem 3 pulls both ways, but the effect due to the comparison on commonly haggled items overwhelms the effect due to the larger set in the poorer society.
4. Discrete effort level sets If effort levels can only be chosen from discrete subsets of R,, the structure of the sets of population equilibria can look much different from simple discrete analogues of what is described in Theorems 1 and 2. To see this, consider the example where W= 2, T= 1, and the admissible effort levels are 0.3, 1.1, 1.5, and 1.7. One population equilibrium has probability masses of 0.4, 0.4, 0, and 0.2 for the respective effort levels and equilibrium rt ~0.1. Another equilibrium has effort-level frequencies of 0.5, 0.3, 0.1, and 0.1, and n=0.2. For another example, let W and T be as above and the admissible effort levels be only 0 and 1. Here all possible probability distributions are population equilibria with 7c equal to the probability mass placed on the 0 effort level. Since population equilibria are the same as symmetric Nash equilibria of certain two-player games, we can expect an approximation result to hold when the discrete effort-level grid gets close in an appropriate sense to the continuum [e.g., Fudenberg and Levine (1986)]. Such general results typically involve s-equilibria, however. Approximating with the uniformly spaced effort-level grid (0, l/n, 2/n,. . . } with n sufficiently large and nW an integer, however, it is easy to see that one exact population equilibrium of discretized version of r places probability mass l/nW on the first nW points of this grid, and yields n = TInW. This equilibrium i.e. the set {0,1/n ,..., W-(1/n)}, construction converges in an appropriate sense to the equilibrium of Theorem 1. Similar approximating constructions are possible for discretized Tu. Uniqueness of the approximating equilibrium is not assured in general,
R. W. Rosenthal, Bargaining rules of thumb
21
however, even when n is large. For example, in discretized r if W = 2 T and W is an even grid point, another population equilibrium places probability mass 2/nW on each element of the odd grid point set {l/n, 3/n,. . . , W-( l/n)}. This equilibrium has rc= 0.
5. Discussion
A. Comparative statics Some cautions against taking the comparative-statics exercise of Section 3 too seriously have been raised already. Here are a few issues that might cause additional problems with an empirical test of claims (i), (ii), and (iii). There are many dimensions on which societies differ that might be correlated with both societal wealth and haggling frequency. For instance, transactions between individuals and governments are rarely haggleable, and different societies have different amounts of government presence in their economies. Furthermore, transactions involving a government seem unlikely to be on average of the same value as those not involving the government. It might therefore be difficult to compare haggling intensities empirically across societies with different-size government sectors. Similarly, it is inherently easier for a large corporation to commit to a policy of fixed prices than it is for a sole proprietorship. Hence it might be difficult to compare haggling intensities across societies in which the typical size of businesses differ. What if T is not a constant fraction of W? In this case, comparative statics involve changing two parameters, and, although I have not attempted it, the analysis and results are bound to be more complicated though in principle no more difficult than that of Theorem 3. For the case when one society has a higher W and a lower T than the other, interpreting the difference between the two societies is problematic. B. The general model Some modifications of the model that seem manageable: l
Allow for the possibility that individuals maintain more than one effort level for use and subdivide the universe of bargaining games into subsets according to which rule is to be used. See Rosenthal (1993) for some ideas on one way to approach this.
l
Allow for more than two outcomes in bargaining. For instance, if one player’s effort level is greater by some margin than the other’s, then his prize is worth some amount more than W and his opponent’s somewhat less than zero (resealing for interpretative purposes if desired).
l
Alter the payoffs so that the winner only expends the same effort as the
22
R. W. Rosenthal,
Bargaining
rules of thumb
loser, even though his effort-level choice is greater. This would allow a more straightforward interpretation of effort level as time spent, but it would eliminate the aspect of constant per-period rule cost from the model and would thereby undermine the rule-of-thumb interpretation. Population equilibrium somehow does not seen as reasonable in this context. More troublesome issues for this approach: 0
It seems clear that informational disparities are an important aspect in many bargaining situations, but it is not obvious how to extend this model in a useful way to accommodate uncertainty.
0
Endogenous choices of both opponents and bargaining games are likely not insignificant in reality, but such effects are difficult to capture in a random-matching model.
0
The analysis of disequilibrium dynamics and stability of equilibria are important aspects that should be addressed in the context of any serious application of the theory. From the brief treatment of these in Rosenthal (1993), it would appear that they will not be easy in these models.
C. Final remarks 0
The ability of individuals to commit not to haggle may itself be a function of choices made by the society. For instance, laws may impose greater or lesser transactions costs, which might in turn affect the relative profitability of sole proprietorships; and, as already noted, sole proprietorships are probably less able to commit to a no-haggle policy. What are the forces that drive such social choices? Do evolutionary forces acting at the societal level select institutions while evolutionary forces at the individual level adapt to them?
0
Unlike most contributions to both the cooperativeand noncooperativegame bargaining literatures, the equilibria of Theorems 1 and 2 admit much concerning behavior in single bargaining indeterminacy encounters. This might be viewed as a strength of this approach by those who regard bargaining as a largely unpredictable activity.
Appendix Results like Theorem Riley (1989)]. I develop
1 have appeared in the literature [cf. Hillman a proof here for the reader’s convenience.
and
Lemma I. There can be no mass points in any population equilibrium of I-. For r,, there can be mass points only at u.
R. W. Rosenthal, Bargaining rules of thumb
Proof.
If in either game there is a mass point of the population
23
equilibrium
F at x # u, for E> 0 sufficiently small rcIF(x+ E)> z~(x) since W > T, contradict-
ing the definition of population equilibrium. Lemma 2. There can be no gaps in the support of any population equilibrium of r. For r,, the only gap can be an open interval the top of which is a mass point at u. Proof.
For any other interval gap (x’,x”), neither x’ nor x” can be a mass point of the equilibrium F by Lemma 1; but then ~~(x’)>rr~(x”), a contradiction. Lemma 3. In any population equilibrium F of r, the bottom of the support of F must be at 0. In T,, the bottom of the support must be either at 0 or at u. In both cases, when the bottom of the support of F is at 0, ~~(0) = 0. Proof. If the bottom of the support of F is x’>O, and x’ is not a mass point of F, then rcn,(x’)
I. From the three lemmas, the c.d.f. F defining any population equilibrium of r is continuous with support being an interval having a bottom at 0. Since rcF(0)=O, every x in the support must satisfy -x+ WF(x)=O. But there is only one c.d.f. with these properties: the uniform c.d.f. on [0, W]. Proof of Theorem 2. (1) If Wju, the population equilibrium for r is obviously an equilibrium for TU. If there is another population equilibrium F for r., it must have a mass point at u; but Q(U) <05rc,(O), a contradiction. If T-C u < W, the population equilibrium of (1) does not work; the (2) only other possibilities involve mass points at U. Placing all the mass at u produces rcn,(u)
or M=(W-U)/(W-T);
and the density on the continuous part can only be the uniform from (1) truncated at W(u - T)/( W- T). If us T, placing all the mass at u obviously generates a population (3) equilibrium. If there is another, its support must contain 0 with rcF(0)= 0. But this is impossible since rcF(u)> 0, unless u = T and all the mass of F is at T, in which case the same equilibrium has been reproduced.
24
Proof of Theorem 3. Denote the equilibrium to consider.
R.W. Rosenthal,
Bargaining rules of thumb
Consider changes from W to W’, where W-c W’. c.d.f.‘s F and F’, respectively. There are several cases
(a) In r or in r, with W’su, both F and F’ are uniform, and F dominates F. (b) If Ws us W’, F is uniform on [0, W], while F’ begins either as a uniform dominating F (when CXW’cu) or with zero density (when u 5 CIW’). In both cases, F’ stays below F on [0, W]. (c) If aW’< u < W, for the initial support segments that are common to both F and F’, F’ dominates F as in (a). Next, W(u - a!)/( W- txW) decreases as a function of W, so F’ remains below F until U. (d) If CXW K u 5 o!W’, all of the mass of F’ is at u, while some of the mass of F is allocated below u. (e) If u saw, F and F’ coincide. References Frank, Robert H., 1988, Passions within reason (Norton, New York). Fudenberg, Drew and David Levine, 1986, Limit games and limit equilibria, Journal of Economic Theory 38, 261-279. Harsanyi, John C. and Reinhard Selten, 1972, A generalized Nash solution for two-person bargaining games with incomplete information, Management Science 18, PS&P106. Hillman, Arye and John Riley, 1989, Politically contestable rents and transfers, Economics and Politics 1, 1740. Leininger, Wolfgang, 1989, Escalation and cooperation in conflict situations - the dollar auction revisited, Journal of Conflict Resolution 32, 231-254. Milgrom, Paul and Robert Weber, 1985, Distributional strategies for games with incomplete information, Mathematics of Operations Research 10, 619-632. Nash, John, 1950, The bargaining problem, Econometrica 28, 155-162. Riley, John and Richard Zeckhauser, 1983, Optimal selling strategies: When to haggle, when to hold firm, Quarterly Journal of Economics 47, 267-289. Rosenthal, Robert W., 1993, Rules of thumb in games, Journal of Economic Behavior and Organization 22, no. 1. Rosenthal, Robert W. and Henry Landau, 1979, A game-theoretic analysis of bargaining with reputations, Journal of Mathematical Psychology 20, 233-255. Rosenthal, Robert W. and Henry Landau, 1981, Repeated bargaining with opportunities for learning, Journal of Mathematical Sociology 8, 61-74. Roth, Alvin E., 1979, Axiomatic models of bargaining (Springer-Verlag, Berlin). Rubinstein, Ariel, 1982, Perfect equilibrium in a bargaining model, Econometrica 50, 97-109. Shubik, Martin, 1971, The dollar auction game: A paradox in noncooperative behavior and escalation, Journal of Conflict Resolution 15, 545-547. Stahl, Ingolf, 1972, Bargaining theory (Economics Research Institute, Stockholm School of Economics, Stockholm). Wang, Ruqu, 1991, Auctions versus posted-price selling, Working paper.
of Economic
Behavior
and Organization
22 (1993)
15-24.
North-Holland
Bargaining rules of thumb Robert
W. Rosenthal*
Boston University, Boston, MA 02215, USA Received
May 1992, final version
received
June 1992
A steady-state, random-matching game model is used to described bargaining in a large population, where bargainers choose rules of thumb to apply across the board in a class of bargaining situations they face. The unique population equilibrium rule distributions are computed for certain cases of the model. These are then used in an exercise to analyze the relationship between a society’s wealth and the relative frequency of haggled transactions in the society.
1. Introduction Within game theory there are two main categories of bargaining theories. Axiomatic bargaining theory emanates from Nash (1950). A typical contribution to this literature’ assumes there is a mapping on some domain of bargaining games that selects the outcome of each game in the domain; it then characterizes the mapping axiomatically. The typical axiom system prescribes both what might constitute reasonable behavior in certain of the games and what might constitute consistency of the mapping across games in the domain. Noncooperative bargaining theory is the second category. It received its main impetus from Rubinstein (1982), although precursors include Harsanyi and Selten (1972) and Stahl (1972). This literature is single-game oriented: a typical contribution posits precise rules for the bargaining process and analyzes Nash or Bayesian equilibria (or refinements thereof) of the resulting game. This paper falls outside both of the categories above and follows the general approach of Rosenthal (1993). It assumes that players face many bargaining situations over time and adopt rules of thumb to apply in all bargaining situations, rather than respond in equilibrium fashion to each single game as it occurs. Thus this theory might be seen as applying to Correspondence to: R.W. Rosenthal, Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA. *Thanks to Jonathan Eaton and Wolfgang Leininger for helpful comments. This research has been supported by National Science Foundation grant no. SES-9010246. ‘Roth (1979) contains an excellent survey of this literature through the late 1970’s. 0167-2681/93/$06.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
16
R. W. Rosenthal,
Bargaining
rules of thumb
frequently-occurring, low-value bargaining situations instead of to an isolated situation in which there is a great deal at stake. It therefore shares with the axiomatic literature a concern for consistency across different games. That concern is motivated here, however, by a belief that much behavior (and in particular much bargaining behavior) is of the automatic-pilot variety and that people choose rules of thumb to apply in broad classes of situations, rather than tine tune behavior to each situation. Like the noncooperative bargaining literature, the theory has a noncooperative-equilibrium type solution concept, termed a population equilibrium. The model is of the largesociety, random-matching variety;2 so the equilibrium conditions are not Nash noncooperative equilibrium conditions, except that they are equivalent to the Nash equilibrium conditions for a particular related game. Unlike the noncooperative bargaining literature, the bargaining process is modeled here in a crude way, with no attention given to specific bargaining procedures. The model is simple enough that it is possible to describe explicitly its unique population equilibrium (at least in the versions of the model in which there are continua of choices) as a function of certain parameters. This means that it is possible to compare equilibria across societies having different characteristics. As an illustration, the model is used here to address the question of why haggling is more prevalent or intense in certain societies and in certain situations than in others. (Others, for example Riley and Zeckhauser (1983) and Wang (1991), have compared haggling with postedprice selling, but the points-of-view, models, and objectives of those studies are different from this one.) The continuum-of-actions versions of the model are spelled out in section 2, and the unique population equilbria are identified for each of these versions of the model. Section 3 contains a comparative-statics exercise, together with interpretive remarks, in which the equilibrium of a relatively poor society is compared with that of a relatively rich one. In section 4, the complications introduced by dropping the continuum-of-actions assumption are discussed. Section 5 consists of a collection of remarks about the model, the results, and interpretations. The proofs of all the results are confined to an appendix.
2. The model and its equilibria Imagine a society consisting of a large number of individuals who are randomly matched in pairs on a regular basis to play bargaining games. The games are all of the following form: Each bargainer simultaneously selects a ‘Other large-society, random-matching models of bargaining Landau (1979) and (1981), but they are considerably different details.
can be found in Rosenthal from this one in both spirit
and and
R.W. Rosenthal, Bargaining rules ofthumb
17
bargaining effort levek3 whoever’s effort level is greater wins the greater share of the object being bargained over. More precisely, in version Z” of the model each matched pair of players plays a random game k, where k indexes parameters (W,, T,J, drawn from a set K of games according to a known distribution (pJkeK. Informed of k,4 the players, call them i and j, simultaneously select effort levels ei and ej from the nonnegative real numbers R +. If e, > ej, i’s payoff is W, -e, and j’s payoff is -ej (and vice versa when i and j are reversed). In the event of a tie, both receive Tk -e,. (Of course, W, > Tk > 0, Vk E K .) Before finishing the description of the model, two aspects of this formulation require immediate emphasis. First, there are only two possible outcomes to any game: win-loss and tie; so the bargaining is indeed of an extremely crude form.’ Second, the winner pays his actual effort; so it is not exactly right to interpret an effort level as the maximum time cost the player is prepared to incur, as in the ‘war of attrition’ [e.g., Milgrom and Weber (1985)] or money cost, as in the ‘dollar auction’ [e.g., Shubik (1971), Leininger (1989)]. Here the players pay their respective effort costs independently of those chosen by their opponents. Thus the individual games resemble a sealed-bid version of the dollar auction. The idea is that players must prepare themselves beforehand for their planned efforts, and that is when the cost is incurred. (See below and section 5 for more on this.) There is assumed to be anonymity and independence in the matching process but enough information about population averages that individuals all assume their current opponent in every period is simply a new random draw from a known distribution of behaviors, the same distribution in each period. Equilibrium requires this distribution to be the true frequency distribution of the population’s actions. Finally, individuals are assumed to choose their effort levels once and for all, and not as a function of k. This is the ‘automatic-pilot’ or rule-of-thumb restriction. For justification, think of effort-level selections as attitudes or emotions that are difficult or impossible for an individual to change, as in Frank (1988). A tough bargainer is what one is or is trained to be, not what one chooses to be. In this interpretation, the force that drives the population toward equilibrium (or at least away from disequilibrium) is an evolutionary one: successful effort levels reproduce; others die out. For further justification, see Rosenthal (1993) and references therein. The effort cost should therefore be interpreted either as a per-game amortization of the one-time effort selected or the flow cost of maintaining the chosen rule. A population equilibrium of r is then a cumulative
3The word ‘effort’ should be taken as a catchall standing for different types of activities that are costly to the player but that increase the chances for success in bargaining. %Cs turns out to be irrelevant in the light of what is to come. sSection 5 revisits this assumption. Adopting 0 as the value of a loss is for expository convenience.
R.W. Rosenthal, Bargaining rules of thumb
18
distribution maximizer
function of:
ACNE
(c.d.f.) F on R +, each element
-x+WF(x)-(W-T)(F(x)-F(x-))
of whose
support
is a
VXER+,
where W -xkpk W, and T=x,p,T,. From this definition6 it should be obvious that the population equilibria of f are identical to the symmetric Nash equilibria of the symmetric twoperson one-shot game in which each player’s pure-strategy space is R, and Player l’s payoff as a function of his strategy x and 2’s strategy y is: (W-x)
if x>y,
(T-x)
Player 2’s payoff is defined Despite the discontinuity is easy to find.
if x=y,
and (-x)
by symmetry. on the diagonal,
Theorem 1. The unique population distribution on [0, W].
equilibrium
if x
the set of population
equilibria
of r is the c.d.f. of the uniform
It might be the case that exogenous constraints limit the amount of effort a player can devote to bargaining; for instance, if effort is for some reason unproductive beyond some point. We shall therefore consider the following variant of r. For the number u>O, define rU to be f with the modification that only the effort levels in [0, u] are permitted, and modify the definition of the population equilibrium in the obvious way. Theorem 2. as follows.
For each u>O, there is a unique population
(1) For u >=W, F is the c.d.f. of the uniform distribution (2) For T
equilibrium
F of T,,
on [0, W];
(xl W) for x < W(u - T)/( W - T) F(x)=
(u-T)/(W-T)
for W(u-T)/(W-T)Sx
1 forxzu (3)
For u5 T,
F(x) =
0 forx
6As c.d.f.‘s are right-continuous, assigned by F to x.
F(x)-F(x-)
simply
denotes
the probability
mass
(if any)
R. W. Rosenthal, Bargaining rules of thumb
19
Notice that in r and in the first two cases of f. the effort of each individual at equilibrium just equals his expected return from the average bargaining game. In case (3) of T,, when u < T, the upper bound on effort levels is so low that even when everyone chooses the largest possible effort level, the gains from bargaining are not exhausted on average. 3. Comparative statics The cross-societal question is: what happens to the population equilibrium when W and T either increase or decrease in size relative to the effort-level scale. To put this question in its simplest form, let us assume that T is a constant fraction TVof W (O
are based on extremely casual observation and hearsay. I have not found literature that bears on cross-society haggling frequency comparisons.
any formal
20
R. W. Rosenthal, Bargaining rules of thumb
societies a larger set of transactions is customarily haggled over (rather than sold by posted prices); (ii) the additional haggled transactions of the poorer society are at the low-value end of the spectrum; and (iii) more effort (time) is expended in haggling over the commonly haggled transactions in the poorer society. Assuming purchasing-power parity for the haggled transactions, that excess supply (and immobility) of labor should hold down the relative value of effort on average in the poorer society, and ignoring both the caveats above and (i) and (ii), the conclusion of Theorem 3 agrees with stylized fact (iii): the same object is valued relatively more highly in the poorer society, so we should expect to see more effort expended in bargaining there. On the other hand, the effects of both (i) and (ii) are to raise the average relative value of the objects haggled over in the richer society; and these effects should increase average haggling in equilibrium in the richer society, again as a consequence of Theorem 3. So, if all three stylized facts are true, our story is that the logic of Theorem 3 pulls both ways, but the effect due to the comparison on commonly haggled items overwhelms the effect due to the larger set in the poorer society.
4. Discrete effort level sets If effort levels can only be chosen from discrete subsets of R,, the structure of the sets of population equilibria can look much different from simple discrete analogues of what is described in Theorems 1 and 2. To see this, consider the example where W= 2, T= 1, and the admissible effort levels are 0.3, 1.1, 1.5, and 1.7. One population equilibrium has probability masses of 0.4, 0.4, 0, and 0.2 for the respective effort levels and equilibrium rt ~0.1. Another equilibrium has effort-level frequencies of 0.5, 0.3, 0.1, and 0.1, and n=0.2. For another example, let W and T be as above and the admissible effort levels be only 0 and 1. Here all possible probability distributions are population equilibria with 7c equal to the probability mass placed on the 0 effort level. Since population equilibria are the same as symmetric Nash equilibria of certain two-player games, we can expect an approximation result to hold when the discrete effort-level grid gets close in an appropriate sense to the continuum [e.g., Fudenberg and Levine (1986)]. Such general results typically involve s-equilibria, however. Approximating with the uniformly spaced effort-level grid (0, l/n, 2/n,. . . } with n sufficiently large and nW an integer, however, it is easy to see that one exact population equilibrium of discretized version of r places probability mass l/nW on the first nW points of this grid, and yields n = TInW. This equilibrium i.e. the set {0,1/n ,..., W-(1/n)}, construction converges in an appropriate sense to the equilibrium of Theorem 1. Similar approximating constructions are possible for discretized Tu. Uniqueness of the approximating equilibrium is not assured in general,
R. W. Rosenthal, Bargaining rules of thumb
21
however, even when n is large. For example, in discretized r if W = 2 T and W is an even grid point, another population equilibrium places probability mass 2/nW on each element of the odd grid point set {l/n, 3/n,. . . , W-( l/n)}. This equilibrium has rc= 0.
5. Discussion
A. Comparative statics Some cautions against taking the comparative-statics exercise of Section 3 too seriously have been raised already. Here are a few issues that might cause additional problems with an empirical test of claims (i), (ii), and (iii). There are many dimensions on which societies differ that might be correlated with both societal wealth and haggling frequency. For instance, transactions between individuals and governments are rarely haggleable, and different societies have different amounts of government presence in their economies. Furthermore, transactions involving a government seem unlikely to be on average of the same value as those not involving the government. It might therefore be difficult to compare haggling intensities empirically across societies with different-size government sectors. Similarly, it is inherently easier for a large corporation to commit to a policy of fixed prices than it is for a sole proprietorship. Hence it might be difficult to compare haggling intensities across societies in which the typical size of businesses differ. What if T is not a constant fraction of W? In this case, comparative statics involve changing two parameters, and, although I have not attempted it, the analysis and results are bound to be more complicated though in principle no more difficult than that of Theorem 3. For the case when one society has a higher W and a lower T than the other, interpreting the difference between the two societies is problematic. B. The general model Some modifications of the model that seem manageable: l
Allow for the possibility that individuals maintain more than one effort level for use and subdivide the universe of bargaining games into subsets according to which rule is to be used. See Rosenthal (1993) for some ideas on one way to approach this.
l
Allow for more than two outcomes in bargaining. For instance, if one player’s effort level is greater by some margin than the other’s, then his prize is worth some amount more than W and his opponent’s somewhat less than zero (resealing for interpretative purposes if desired).
l
Alter the payoffs so that the winner only expends the same effort as the
22
R. W. Rosenthal,
Bargaining
rules of thumb
loser, even though his effort-level choice is greater. This would allow a more straightforward interpretation of effort level as time spent, but it would eliminate the aspect of constant per-period rule cost from the model and would thereby undermine the rule-of-thumb interpretation. Population equilibrium somehow does not seen as reasonable in this context. More troublesome issues for this approach: 0
It seems clear that informational disparities are an important aspect in many bargaining situations, but it is not obvious how to extend this model in a useful way to accommodate uncertainty.
0
Endogenous choices of both opponents and bargaining games are likely not insignificant in reality, but such effects are difficult to capture in a random-matching model.
0
The analysis of disequilibrium dynamics and stability of equilibria are important aspects that should be addressed in the context of any serious application of the theory. From the brief treatment of these in Rosenthal (1993), it would appear that they will not be easy in these models.
C. Final remarks 0
The ability of individuals to commit not to haggle may itself be a function of choices made by the society. For instance, laws may impose greater or lesser transactions costs, which might in turn affect the relative profitability of sole proprietorships; and, as already noted, sole proprietorships are probably less able to commit to a no-haggle policy. What are the forces that drive such social choices? Do evolutionary forces acting at the societal level select institutions while evolutionary forces at the individual level adapt to them?
0
Unlike most contributions to both the cooperativeand noncooperativegame bargaining literatures, the equilibria of Theorems 1 and 2 admit much concerning behavior in single bargaining indeterminacy encounters. This might be viewed as a strength of this approach by those who regard bargaining as a largely unpredictable activity.
Appendix Results like Theorem Riley (1989)]. I develop
1 have appeared in the literature [cf. Hillman a proof here for the reader’s convenience.
and
Lemma I. There can be no mass points in any population equilibrium of I-. For r,, there can be mass points only at u.
R. W. Rosenthal, Bargaining rules of thumb
Proof.
If in either game there is a mass point of the population
23
equilibrium
F at x # u, for E> 0 sufficiently small rcIF(x+ E)> z~(x) since W > T, contradict-
ing the definition of population equilibrium. Lemma 2. There can be no gaps in the support of any population equilibrium of r. For r,, the only gap can be an open interval the top of which is a mass point at u. Proof.
For any other interval gap (x’,x”), neither x’ nor x” can be a mass point of the equilibrium F by Lemma 1; but then ~~(x’)>rr~(x”), a contradiction. Lemma 3. In any population equilibrium F of r, the bottom of the support of F must be at 0. In T,, the bottom of the support must be either at 0 or at u. In both cases, when the bottom of the support of F is at 0, ~~(0) = 0. Proof. If the bottom of the support of F is x’>O, and x’ is not a mass point of F, then rcn,(x’)
I. From the three lemmas, the c.d.f. F defining any population equilibrium of r is continuous with support being an interval having a bottom at 0. Since rcF(0)=O, every x in the support must satisfy -x+ WF(x)=O. But there is only one c.d.f. with these properties: the uniform c.d.f. on [0, W]. Proof of Theorem 2. (1) If Wju, the population equilibrium for r is obviously an equilibrium for TU. If there is another population equilibrium F for r., it must have a mass point at u; but Q(U) <05rc,(O), a contradiction. If T-C u < W, the population equilibrium of (1) does not work; the (2) only other possibilities involve mass points at U. Placing all the mass at u produces rcn,(u)
or M=(W-U)/(W-T);
and the density on the continuous part can only be the uniform from (1) truncated at W(u - T)/( W- T). If us T, placing all the mass at u obviously generates a population (3) equilibrium. If there is another, its support must contain 0 with rcF(0)= 0. But this is impossible since rcF(u)> 0, unless u = T and all the mass of F is at T, in which case the same equilibrium has been reproduced.
24
Proof of Theorem 3. Denote the equilibrium to consider.
R.W. Rosenthal,
Bargaining rules of thumb
Consider changes from W to W’, where W-c W’. c.d.f.‘s F and F’, respectively. There are several cases
(a) In r or in r, with W’su, both F and F’ are uniform, and F dominates F. (b) If Ws us W’, F is uniform on [0, W], while F’ begins either as a uniform dominating F (when CXW’cu) or with zero density (when u 5 CIW’). In both cases, F’ stays below F on [0, W]. (c) If aW’< u < W, for the initial support segments that are common to both F and F’, F’ dominates F as in (a). Next, W(u - a!)/( W- txW) decreases as a function of W, so F’ remains below F until U. (d) If CXW K u 5 o!W’, all of the mass of F’ is at u, while some of the mass of F is allocated below u. (e) If u saw, F and F’ coincide. References Frank, Robert H., 1988, Passions within reason (Norton, New York). Fudenberg, Drew and David Levine, 1986, Limit games and limit equilibria, Journal of Economic Theory 38, 261-279. Harsanyi, John C. and Reinhard Selten, 1972, A generalized Nash solution for two-person bargaining games with incomplete information, Management Science 18, PS&P106. Hillman, Arye and John Riley, 1989, Politically contestable rents and transfers, Economics and Politics 1, 1740. Leininger, Wolfgang, 1989, Escalation and cooperation in conflict situations - the dollar auction revisited, Journal of Conflict Resolution 32, 231-254. Milgrom, Paul and Robert Weber, 1985, Distributional strategies for games with incomplete information, Mathematics of Operations Research 10, 619-632. Nash, John, 1950, The bargaining problem, Econometrica 28, 155-162. Riley, John and Richard Zeckhauser, 1983, Optimal selling strategies: When to haggle, when to hold firm, Quarterly Journal of Economics 47, 267-289. Rosenthal, Robert W., 1993, Rules of thumb in games, Journal of Economic Behavior and Organization 22, no. 1. Rosenthal, Robert W. and Henry Landau, 1979, A game-theoretic analysis of bargaining with reputations, Journal of Mathematical Psychology 20, 233-255. Rosenthal, Robert W. and Henry Landau, 1981, Repeated bargaining with opportunities for learning, Journal of Mathematical Sociology 8, 61-74. Roth, Alvin E., 1979, Axiomatic models of bargaining (Springer-Verlag, Berlin). Rubinstein, Ariel, 1982, Perfect equilibrium in a bargaining model, Econometrica 50, 97-109. Shubik, Martin, 1971, The dollar auction game: A paradox in noncooperative behavior and escalation, Journal of Conflict Resolution 15, 545-547. Stahl, Ingolf, 1972, Bargaining theory (Economics Research Institute, Stockholm School of Economics, Stockholm). Wang, Ruqu, 1991, Auctions versus posted-price selling, Working paper.