Possibly Indecisive Contests, where the prize may be not awarded. In the latter, any ... 3In this paper we confine ourselves to pure strategies. 4See Acemoglu ...
Properties of contests Luis Corchón and Marco Serenay First draft: November, 2013 This draft: November, 20th, 2016
Abstract We aim at characterizing which kind of functions can be rationalized as the best reply of payo¤ maximizing agents in contests for a …xed prize. We show that the rationalizability strongly di¤er between Decisive Contests, where the prize is allocated with certainty, and Possibly Indecisive Contests, where the prize may be not awarded. In the latter, any arbitrary set of best reply functions is rationalizable, thus "anything goes". In the former, on the contrary, the best reply functions have to satisfy strong conditions. JEL Classi…cation: C62, C72
1
Introduction
The shape of the best reply function is crucial to address uniqueness, stability and comparative statics of equilibria in normal form games.1 In general, the best reply could be any function. But in particular games, one would expect that best replies satisfy additional properties. In this paper we consider a very special kind of game, a contest, and investigate the kind of best replies that may arise. Recall that in a contest, the payo¤ function of each player takes a very special form: it is linear in expected revenue minus costs and Department of Economics, Universidad Carlos III, Madrid. Max Planck Institute for Tax Law and Public Finance, Munich. 1 This question …rst arose in general equilibrium. Sonneneschein (1972) (re…ned by Mantel, 1974, and Debreu, 1974) showed that there are no structural properties of excess demand functions beyond continuity, a boundary property and Walras Law. y
1
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the cost is linear in the expense made by the player.2 Moreover we assume that marginal costs and prize valuations are identical for all individuals. Formally, given a number V (which will be used as the valuation of the player), a list of n functions is rationalizable when there are n payo¤ functions linear in expected revenue minus costs, such that, when maximized, yield a unique best reply identical to these functions.3 Essential to rationalizability is the construction of a Contest Success Function (CSF) which maps e¤ort of players into the probability of getting the prize. In addition to be a probability, and in order to tie our hands as tightly as possible, we require this functions to be di¤erentiable, increasing in the e¤ort of the contestant, decreasing in the e¤ort of everybody else and strictly concave. We distinguish between two types of contests. In a Decisive Contest (DC) the sum of the probabilities is equal to one. For instance, in contests like the World Cup (any match in a Cup actually) or the allocation of Olympic games, there must be a winner. In a Possibly Indecisive Contest (PIC) the prize may be not awarded. For instance, some literary or scienti…c prizes may be declared vacant, in R&D contests the invention may not be made, some wars end up in an impasse, some trials produce a hung jury, some vacancies are not …lled, etc. Note that DCs are a subset of PICs. We …nd that there is an important di¤erence between DC and PIC regarding the best reply functions that can be rationalized. We show that in a PIC, anything goes. In particular, any continuous function satisfying some mild properties can be rationalized. In other words, the (apparently strong) assumptions made on the form of the payo¤ function and the CSF are not su¢ cient to give any structure to the problem. This negative result casts the shadow of a doubt on the endeavor of …nding unique and stable equilibria in this model. Unless other properties are added, our result shows that there is a lot of indeterminacy on the theoretical predictions of this kind of models.4 These properties may be inspired from the research done in the microfoundations of CSF.5 In other words, our negative results plus the previous …ndings 2
We only consider here contests with a unique prize. In this paper we con…ne ourselves to pure strategies. 4 See Acemoglu and Jenssen (2013), section 5.2 for an analysis of PIC under the assumption that the CSF takes a special form. 5 These models include random e¤ort models (Dixit, 1987; Hillman and Riley, 1989; Fullerton and McA¤ee, 1999), axiomatics (Skaperdas, 1996; Clark and Riis, 1998) or contests organized by a planner (Epstein and Nitzan, 2006; Corchón and Dahm, 2010 and 2011). 3
2
Electronic copy available at: https://ssrn.com/abstract=2882537
may be useful to shape future research. The picture is considerably more complicated in DC contests. In the twoplayer case, we show that if a pair of functions is rationalizable, the slopes in the intersection must be of di¤erent sign. When these two functions cross in the diagonal (i.e., in any symmetric Nash equilibrium) their slopes must be zero.6 This implies that the symmetric Nash equilibrium of any 2-person contest is unique. This …nding has an implication on the n-player case, namely, under the assumption that the rationalizing CSF attaches a constant probability to a player exerting zero e¤ort, we show that the property of slopes of opposite sign holds for any pair of contestants when the other contestants exert zero e¤ort. To end this part, we consider a local notion of rationalizability in the n-player case, in which the properties of the CSF hold only in a neighborhood of the intersection of the considered functions. We prove that any list of decreasing functions is locally rationalizable and show that in some cases this result leads to full rationalization. The requirement that the rationalizable functions must be decreasing comes from the fact that these functions necessarily appear in the CSF as the part that depends on other contestants’strategies. If we drop the requirement that the probability that contestant i obtains the prize must be decreasing in other contestants’e¤orts, then any set of functions is locally rationalizable. Our results can be compared with those obtained in Cournot oligopoly. Both Cournot and contests share several common features: players (…rms, or contestants) simultaneously choose actions (quantities, or e¤orts), and the "cake" won by each player (inverse demand function, or CSF) depends on every player’s action. In Cournot, the overall size of the cake at stake is not …xed, like in PIC; in fact, our "anything goes" result in PIC is the correlate of the "anything goes" result found by Corchón and Mas-Colell (1996) in Cournot oligopoly, although the two results are not logically related and the restrictions imposed in these two classes of models are naturally di¤erent.7 In DC the fact that CSF is a probability explains our results for two contestants. Furth (2009) shows that not everything goes, The common ground is that imposing restrictions jeopardizes the anything goes result. In particular, 6
This explains the curious fact that best replies derived from speci…c CSF, i.e. ratio (Tullock, 1980), additive or logit (Dixit, 1987), egalitarian-relative e¤ort (Nitzan, 1991), noisy ratio (Amegashie, 2006), additive (Polishchuk and Tonis, 2013), relative di¤erence (Beviá and Corchón, 2015), all have the property that in a symmetric equilibrium with two players the best reply has zero slope. 7 In fact, our and their construction are based on completely di¤erent ideas.
3
imposing homogeneity of the oligopoly,and imposing DC (that is, …xed overall cake), we show that not everything goes. We hope that our work stimulates research on the properties of contests beyond those that can be obtained by computing the equilibrium or by looking at the …rst order conditions of payo¤ maximization.
2
Model of contest
Consider a contest among n contestants, with n an agent whose payo¤ function is i (x)
= pi (x1 ; :::; xn )V
2. A contestant, say i, is xi
(1)
where V > 0 is the prize valuation, which is identical across contestants, and pi (x1 ; :::; xn ) is the contest success function (CSF). The best reply of contestant i is a function xi = i (x i ) in which x i = (x1 ; :::; xi 1 ; xi+1 ; :::; xn ) such that i ( i (x i ); x i ) i (xi :x i ) for all xi . We require the CSF to satisfy some properties. Namely, it is a wellbehaved probability and that together the probabilities do not sum up to more than 1: (Prob) pi (x1 ; :::; xn ) 2 [0; 1] and 8(x1 ; :::; xn ) 2 Rn+
n P
pi (x1 ; :::; xn )
i=1
1, 8i 2 f1; :::; ng and
and, in order to be able to use the …rst-order approach, we require the CSF to be di¤erentiable and strictly concave along the best reply function. Formally, we impose that 8i 2 f1; :::; ng, 8(x1 ; :::; xn ) 2 (0; V ] and when xi = i (x i ), (Di¤) pi (x1 ; :::; xn ) is di¤erentiable8 (Incr)
@pi (x1 ;:::;;xn ) @xi
0
(Decr)
@pi (x1 ;:::;;xn ) @xj
0 8j 2 f1; :::; ng with j 6= i
8
We assume that pi () can be extended to a di¤erentiable function on an open set containing its domain for allowing this to be closed and not open, as the standard de…nition of di¤erentiability requires.
4
(Conc)
@ 2 pi (x1 ;:::;;xn ) @x2i
0, a set of functions f 1 ; :::; n g is rationalizable if for every contestant i there exists a payo¤ function of the form (1) where pi (x1 ; :::; xn ) satis…es (Prob)-(Di¤ )-(Incr)-(Decr)-(Conc) with a prize n P V = K which yields as the unique best reply i (x i ) whenever xj > 0. j6=i
Our de…nition can be understood in terms of two …ctitious agents: one who announces a positive number and a set of functions, and the other who will look for a set of CSF which converts these functions into the best replies of a normal form game de…ned by (1) with a prize corresponding to the announced number. For the …rst result on rationalizability, we impose some regularity assumption on the function to be rationalized.
5
De…nition 3 A function i satis…es (Reg) if [0; V ]n 1 ! [0; Vn ], and 9k < Vn s.t. i (x i ) < k.
i
is di¤erentiable, and
i
:
The restriction of the action space to the interval [0; Vn ] guarantees "aggregate individual rationality", which means that under (Reg) the maximum aggregate cost of e¤ort that the set of best reply functions might prescribe is V . For the special case of Tullock’s best reply, the maximum value reached by i is in the (symmetric) equilibrium, i.e. when players play xi = nn21 V , 8i 2 f1; :::; ng. Note that nn21 V < Vn , and hence in Tullock the maximum value of the best reply function is lower than the one imposed by (Reg). Then we have the following result. Proposition 4 Let f 1 ; :::; n g satisfy (Reg). Then f 1 ; :::; nalizable in Possibly Indecisive Contest.
ng
are ratio-
Proposition 4 has a straightforward implication on rationalization in DC. Consider the following two contests: i) a PIC with n 1 players, and ii) a DC with n players, where the …rst n 1 players have the same CSF as the (n 1) players of the PIC. Then one can add an auxiliary nth -player to the PIC whose CSF coincides with the probability of not awarding the prize. This way, any set of n 1 best reply functions is rationalizable in an n-player DC. Formally, Corollary 5 Let f 1 ; :::; n 1 g satisfy (Reg). Then there is a function such that f 1 ; :::; n 1 ; n g are rationalizable in a DC.
n
It is less trivial to rationalize an arbitrary set of n best reply functions in an n-player DC. This is what we address in the next section.
4
Rationalization in Decisive Contests
Rationalization in PIC relies on the fact that, once a set of arbitrary best reply functions is rationalized by a corresponding set of CSFs, we can cherrypick the probability of not awarding the prize to equal the probability which remains from the sum of all the CSFs to reach 1. Moving from PIC to DC 6
makes us lose such degree of freedom, and thus simultaneous rationalization of a best reply function for each contestant is harder to achieve. Think about the two-contestant case: once 1 is rationalized by a certain p1 , then p2 = 1 p1 , and thus a non-arbitrary 2 necessarily follows. The analysis brings about sharply di¤erent result when n = 2 and when n > 2. For this reason, we treat these two cases separately.
4.1
The case of n = 2
Now we show that the result on rationalizability in PIC (Proposition 4) does not carry over to DC when there are two contestants. In Proposition 6 below we prove that, in this case, not every pair of functions is rationalizable. We say that the slopes of two functions are opposite when the product of both slopes is negative. Proposition 6 Let f 1 ; 2 g be a pair of rationalizable best reply functions. In any point (x1 ; x2 ) such that x1 = 1 (x2 ) and x2 = 2 (x1 ) either the sign of the slopes of 1 and 2 are opposite or are both zero. Proof. We …rst write the FOC of payo¤ maximization. Since p1 + p2 = 1 @p1 (x1 ; x2 ) V1 = 1 and @x1
@p1 (x1 ; x2 ) V2 = 1 @x2
(2)
If x1 = (x2 ) can be rationalized by a CSF p1 ( ), di¤erentiating (2) yields d (x2 ) = dx2
@ 2 p1 (x1 ;x2 ) @x1 @x2 @ 2 p1 (x1 ;x2 ) @x21
d (x1 ) and = dx1
@ 2 p1 (x1 ;x2 ) @x2 @x1 @ 2 p1 (x1 ;x2 ) @x22
:
(3)
And the result follows from the fact that the SOCs hold with strict inequality. Proposition 6 provides strong necessary conditions for an equilibrium when n = 2. It implies that if strategies are strategic substitutes for an agent they must be strategic complements for the other agent (see also Dixit, 1987), or if the two contestants are identical and equilibrium is symmetric, then the slopes of best replies must be zero (as we know it happens with the Tullock CSF). Thus, all functions which do not satisfy these conditions are not rationalizable. A corollary of this result is that if best replies are identical, there is, at most, a unique symmetric equilibrium. 7
Corollary 7 When n = 2 and the best reply functions are identical there is at most one symmetric equilibrium. Proof. Consider a symmetric equilibrium (x ; :::; x ), such that x = (x ). Since all best reply functions are identical, by Proposition 6 we know that d (x ) = 0 8i = 1; :::; n. Assume by contradiction that equilibria are multiple. dxi If there is a continuum of symmetric NE over a set C [0; V ], then it lies along the line xi = xj 8i 6= j, by de…nition of symmetric equilibrium; thus, d (x) (x ) = 1 8x 2 C, which contradicts the equilibrium condition d dx = 0. dxi i If there is a …nite number of equilibria, pick two consecutive ones: namely, (x ; :::; x ) and (x ; :::; x ), with x > x . Then, 8xi 2 (x ; x ) either (x ) = 0, whereas the (xi ) > xi or (xi ) < xi . The former contradicts d dx i latter contradicts
4.2
d (x ) dxi
= 0.
The case of n > 2
Proposition 6 has an implication for the case n > 2. Suppose that we want to rationalize a list of functions f 1 ; 2 ; ::: n g by using a CSF that yields a constant probability to agents making zero e¤ort, i.e. xi = 0 ! pi (x1 ; :::; ; xn ) = ci
0:
(4)
In the case of Tullock CSF, ci = 0 (i.e. "no pain, no gain"), but in the CSF proposed in Nitzan (1991), Grofman and Skaperdas (1995) or Beviá and Corchón (2015), ci may be positive.9 Take any pair of functions j ; k and set xr = 0 for all r 6= j; k.PThis is now like a contest played by contestants j and k in which pj = 1 pk . It is easy to check that Proposition i6=j;k ci 6 carries over to this case and thus, at any point (xj ; xk ) such that xj = j (0; ::; 0; xk ; 0; ::0) and xk = k (0; ::; 0; xj ; 0; ::; 0) the slopes of j ( ) with respect to xk and of k ( ) with respect to xj at (xj ; xk ) must either be both zero or have opposite sign. If a pair of functions fails this test, the list of functions is not rationalizable by a CSF ful…lling property (4). We remark that there are CSF that do not ful…ll this property like those proposed by Baik (1988), Che and Gale (2000) or Amegashie (2006). Thus we should dwell more on the study of lists of functions that are rationalizable when n > 2 and 9
Notice that ci has to be su¢ ciently low so as not to contradict (Prob).
8
the (4) property does not hold.10 Unfortunately, we have to restrict ourselves to a weaker notion of rationalization, namely, "Local Rationalization". De…nition 8 Given K > 0, a set of functions f 1 ; :::; n g satisfying (Reg) is locally rationalizable if there is a list of CSFs fp1 ; :::; pn g such that if (x1 ; :::; xn ) is a …xed point of f 1 ; :::; n g i) fp1 ; :::; pn g satisfy (Prob)-(Di¤ )-(Incr)-(Decr)-(Conc) in a neighborhood of fx1 ; :::; xn g ii) (x1 ; :::; xn ) is a Nash equilibrium of a game with payo¤s function of the form (1) with a prize V = K iii)
@
i (x1 ;:::;xn )
@xj
=
@ 2 pi (x1 ;:::;xn ) @xi @xj @ 2 pi (x1 ;:::;xn ) @ 2 x2 i
for all i 6= j.
The notion of local rationalization does not attempt to track down the functions on the whole domain of de…nition, just in a neighborhood of the i …xed point (whose existence is guaranteed by Brower’s …xed point theorem). When the i functions are linear, local rationalization is just rationalization except that pi may be not globally increasing (resp. decreasing) in xi (resp. xj ; j 6= i). As a preparation for our last result on local rationalization when n > 2, let us write the FOC for players 1; :::; n 1 and for player n. Respectively, they are 1 @pi (x1 ; :::; xn ) = @xi V ! n 1 X @pi (x1 ; :::; xn ) 1 = @xn V i=1 10
i = 1; :::; n
1
(5) (6)
There are many examples of results that are di¤erent in two dimensions and in more than two dimensions. Thus the Poincare-Bendixon theorem on limit sets of di¤erential equations (Hirsch and Smale (1974) p. 239), Arrow impossibility theorem (which needs more than two alternatives), Nash implementation (Corchon (2009) third paragraph after Theorem 7) or statistics (Saari 1987). We will see that our topic adds another example to the list above.
9
And the SOC @ 2 pi (x1 ; :::; xn ) < 0 @x2i n 1 2 X @ pi (x1 ; :::; xn )
@x2n
i=1
i = 1; :::; n
(7)
1
(8)
> 0
We di¤erentiate (14) with respect to xj to obtain the slopes of the best replies, dxi dxj
=
dxn = dxj
@ 2 pi (x1 ;:::;xn ) @xi @xj @ 2 pi (x1 ;:::;xn ) @ 2 x2i nP1 i=1 nP1 i=1
i = 1; :::; n
1; j = 1; :::; n and j 6= i
(9)
@ 2 pi (x1 ;:::;xn ) @xj @xn
j = 1; :::; n
(10)
1
@ 2 pi (x1 ;:::;xn ) @x2n
Note that for any arbitrary point (x1 ; :::; xn ) and given
dpi (x1 ;:::;xn ) dxj
@ 2 pi (x1 ;:::;xn )
i 6= j,
there are i 6= j 6= k such that (9) and (10) hold. @xj @xk Now we are ready to prove our last result. Let i (x i ) with i = 1; :::; n be a list of functions satisfying (Reg). By Brouwer …xed point theorem there is a …xed point (x1 ; :::; xn ). Then: Proposition 9 Let f 1 ; :::; n g be a list of decreasing functions satisfying (Reg). Then f 1 ; :::; n g are locally rationalizable in Decisive Contests. Proof. Let11 pi (x1 ; :::; xn ) =
ai x i +
n n X X
aij;k xj xk
j=1 k=1;k6=j
pn (x1 ; :::; xn ) = 1
n 1 X
1 2 x + an x n 2 i
1 2 x 2 n
!
+
i
pi (x1 ; :::; xn )
i=1
11
Note that the constructed CSF (11) does not satisfy property (4) because when contestant i makes zero e¤ort, pi depends on xj .
10
i = 1; :::; n(11)1
(12)
First note that given that x’s are in a compact set there is a maximum and a minimum of the term that multiplies . This implies that setting su¢ ciently small and choosing i conveniently, pi 2 [0; 1], i = 1; :::; n 1. Thus the candidate in (11) and (12) is indeed a CSF. Note that the values of i ’s do not a¤ect FOC, SOC and the comparative static properties of the contest embodied in the second cross derivatives. Now setting i = pi V xi , i = 1; :::; n 1 we have that FOC of payo¤ maximization are ai +
n X
aii;j xj
xi =
j=1;j6=i
1 V
i = 1; :::; n
1
(13)
2
We see that SOC hold for any value of aij;k ’s because @@xp2i = 1, as assumed i before. Now choose dpi (x1 ; :::; xn ) aij;k = ; j 6= k (14) dxk and ai ; i = 1; :::; n 1 such that the equation (13) holds at (x1 ; :::; xn ). Thus, from the point of view of players 1; :::; n 1, we have obtained local rationalization because the candidate CSF ful…lls all the properties: It is di¤erentiable, increasing in xi , decreasing in xj (because 0 s are decreasing) and strictly concave. Now consider player n, whose payo¤ is ! n 1 X 1 pi (x1 ; :::; xn ) V xn i=1
and thus FOC are n 1 X @pi (x1 ; :::; xn ) i=1
@xn
=
1 V
or equivalently, n X n 1 X
aij;n xj + an
i=1 j=1
xn =
1 V
(15)
Then, SOC hold. Now we have that dpn (x1 ; :::; xn ) @ 2 pn (x1 ; :::; xn ) X i = = aj;n dxj @xn xj i=1 n 1
11
(16)
Compare (14) and (16). In particular, in (16) choose the parameters aij;n whenever i 6= j such that (16) holds. Finally choose an such that the left 1 at (x1 ; :::; xn ). And the proof is complete. hand side of (15) equals V Proposition 9 is not as powerful as we had wished on two counts. First, it assumes that the functions to be rationalized are decreasing. This guarantees that the probability that contestant i obtains the prize is decreasing in the other contestants’ e¤ort. Thus our result leaves open the rationalizability of a very large set of functions. Second, we only obtain local rationalization. If the functions to be rationalized were linear, then our construction would ful…ll all the properties of a rationalizable function except that our CSF does not satisfy (Incr) globally, just locally. This problem can be solved by using a weaker de…nition of rationalizability in which the value of the prize is not given and may be di¤erent for each contestant. In this case, (Incr) is guaranteed by choosing ai in the FOC (13) such that ! n P i ai + min ai;j xj xi > 0 (the minimum exists because e¤orts are j=1;j6=i
in a compact set), and then choosing Vi to ful…ll the FOC. Additionally, if (x1 ; :::; xn ) is not far from the maximum values of e¤orts — see (Reg) — , (Incr) holds globally. Summing up, under additional assumptions an "everything goes" result can be proved when n > 2.
References [1] Acemoglu, D., and Jensen, M. K. (2013). "Aggregate Comparative Statics", Games and Economic Behavior, 81, 27-49. [2] Amegashie, J. A. (2006). ”A contest success function with a tractable noise parameter", Public Choice, 126, 135-144. [3] Beviá, C., and L. C. Corchón (2015). "Relative di¤erence contest success function", Theory and Decision, 78, 377-398. [4] Baik, K. H. ”Di¤erence-Form Contest Success Functions and E¤ort Level in Contests”. European Journal of Political Economy, 14, 685-701, 1998.
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[5] Che, Y.-K. and I. Gale (2000), “Di¤erence-Form Contests and the Robustness of All-Pay Auctions,” Games and Economic Behavior 30, 22– 43. [6] Clarke, D. and C. Riis (1998). "Contest Success Functions: An Extension", Economic Theory, 11, 201-4. [7] Corchón, L. C. (1994). "Comparative statics for aggregative games the strong concavity case", Mathematical Social Sciences, 28, 151-165. [8] Corchón, L. C. "The Theory of Implementation: What Did We Learn?". Encyclopedia of Complexity and Systems, Editor Meyers, Robert A. Springer, May 2009. [9] Corchón, L. C., and M. Dahm (2010). "Foundations for Contest Success Functions", Economic Theory, 43, 81-98. [10] Corchón, L. C., and M. Dahm (2011). "Welfare maximizing contest success functions when the planner cannot commit", Journal of Mathematical Economics, 47, 309-317. [11] Corchón, L. C., and Mas-Colell, A. (1996). "On the stability of best reply and gradient systems with applications to imperfectly competitive models", Economics Letters, 51, 59–65. [12] Debreu, G. (1974). "Excess Demand Functions", Journal of Mathematical Economics, 1, 15-21. [13] Dixit, A. (1987). "Strategic Behavior in Contests", The American Economic Review, 77, 5, 891-898. [14] Epstein, G., and S. Nitzan (2006). "The Politics of Randomness", Social Choice and Welfare, 27, 423-433. [15] Fullerton, R. L., and McAfee, R. P. (1999). "Auctioning Entry into Tournaments", Journal of Political Economy, 107, 573-605. [16] Furth, D. (2009). "Anything goes with heterogeneous, but not always with homogeneous oligopoly", Journal of Economic Dynamics & Control, 33, 183-203.
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[17] Grofman B., and S. Skaperdas (1995). "Modeling negative campaigning." The American Political Science Review, 89, 1, 49-61. [18] Hillman, A., and J. Riley (1989). "Politically Contestable Rents and Transfers", Economics and Politics, 1, 17-39. [19] Mantel, R. (1974). "On the Characterization of Aggregate Demand", Journal of Economic Theory, 7, 348-353. [20] Nitzan, S. (1991). "Collective Rent Dissipation", The Economic Journal 101, 1522-1534. [21] Polishchuk, L., and A. Tonis (2013). "Endogenous Contest Success Functions: a Mechanism Design Approach", Economic Theory, 52, 271-297. [22] Saari, D. (1987). "The Source of Some Paradoxes from Social Choice and Probability". Journal of Economic Theory, 41, 1-22. [23] Skaperdas, S. (1996). "Contest Success Functions", Economic Theory, 7, 283-290. [24] Sonneneschein, H. (1972). "Market excess Demand Functions", Econometrica, 549-563. [25] Tullock, G. (1980). "E¢ cient Rent-Seeking", in J.M. Buchanan, R.D. Tollison and G. Tullock (eds.) Towards a Theory of a Rent-Seeking Society, Texas A&M University Press 97-112.
A
Appendix
Proof of Proposition 4. Let us de…ne the CSF for each contestant as: pi (x) = min
A(x) + (nV V
14
y);
1 n
(17)
where A(x) = y =
i (x i ) n X
+
b y
xi y +1 b
ln
xj
j6=i
x = fx1 ; :::; xn g and b 2 R++ . Note that (17) is not di¤erentiable. However, 8x i the best reply is in the di¤erentiable part; namely, where pi (x) = A(x) + (nV y). V In fact, we require the CSF to be di¤erentiable only along the best reply function — see (Di¤). The min-operator of (17) is needed to guarantee that the CSF satis…es (Prob) in the whole domain, that is, 8x 2 Rn++ , and not only along the best reply function. The proof is structured in three steps. i o nh A(x) + (nV y) V xi = i (x i ). In Step 1, we prove that arg max V xi 0
In Step 2, we prove that p satis…es the …ve standard properties of CSFs spelled out in Section 2. In Step 3, we prove that we can ignore the minoperatornh of (17) when looking for othe best reply function; more formally, i A(x) + (nV y) V xi = arg max fpi (x)V xi g. Thus, Step arg max V xi 0
xi 0
1, Step 2 and Step 3 imply that any i is rationalizable by (17). Since pi (x) 1 by construction, the same procedure can be iterated 8i 2 f1; :::; ng without n n P violating that pi (x) 1, and thus the whole set f 1 ; :::; n g is indeed i=1
rationalizable. Step 1. h The …rst-order condition of the problem max A(x) + (nV V xi 0
1 V
"
b i (x i ) + y xi y +1 b
# y V b
1
= () () () 15
i y) V
0 y b
i (x i ) + xi y +1 b
1
=1
y y i (x i ) + 1 = xi + 1 b b xi = i (x i )
xi is
In order to show that xi = i (x i ) is the unique best reply, we need to check that playing xi = 0 does not give to contestant i greater payo¤ 8x i . i ( i (y); x i )
1 V
i (x i )
+
b y
ln
i (x i )y
b
+ 1 + (nV i (x i )
+
b y
y) V ln ln
i (x i )y
b i (x i )y
b
i (x i )
i (0; x i )
[(nV
+1
y)] V
i (x i ) i (x i )y
b
+1
i (x i )y
b
z and the last inequality holds by the fact that ln(z + 1) z+1 i¤ z > 1. A(x) Uniqueness immediately follows by strict concavity of V + (nV y). Step 2. (Prob) pi (x) 1 holds by construction. pi (x) 0 is equivalent to A(x) + (nV y) 0, which holds because A(x) 0 and y 2 [0; nV ]. V (Incr) and (Conc) trivially hold by the fact that xi enters (17) only in the logarithm contained in A(x), and that by Step 1 we know that along the + (nV y).12 best reply lies pi (x) = A(x) V (Decr) For @ p@xi (x) 0 to hold, it is su¢ cient that the …rst term of the j min-operator in (17) is decreasing in xj .
@ A(x) + (nV @xj V
y)
0 () ()
@[A(x)=V ] @ [(nV y)] + @xj @xj @[A(x)=V ] 1 0 @xj
0 (18)
We now prove that lim @A(x) = 0, which su¢ ces for (18) to hold (b is an @xj b!1
12
Note that, since (Conc) requires strict concavity, it has to hold that y > 0, which is what we impose in the de…nition of rationalizability.
16
+1
arbitrarily large parameter). @A(x) lim = b!1 @xj
b!1
=
lim
= =
lim (
b!1
lim
b xy ) ln( + 1) + lim 2 b!1 y b
0 x (y)
ln( xy + 1) b y2
y2 0 x (y)
+ lim
y2 b!1 y 2 0x (y) b b ( xy ) xy+b b2 lim y2 b!1 (y 2 0x (y) b)2
x (y)
+
x y
+
x y
x b
+ x yb
xy + b
b!1
b
ln( xy + 1) b
x
b x (y) + y xy +1 b
(by L’Hopital’s rule)
2
bxy (y 2 0x (y) b) x + 2 2 b!1 b y (xy + b) y x x = + y y = 0 =
lim
(Di¤) Di¤erentiability of pi follows from the di¤erentiability of i , see (Reg). Step 3. We can ignore the min-operator of (17) when looking for the best reply function. In fact, if the min-operator a¤ected the best reply, then we would + (nV y) > n1 and pi (x) = n1 . But if so, an have that, for some xi , A(x) V arbitrarily small decrease in xi would not a¤ect pi (x) and entail a smaller cost of e¤ort, making it pro…table to decrease xi . Thus, any such xi cannot be part of the best reply.
17
