Axiomatization of Reverse Nested Lottery Contests

Axiomatization of Reverse Nested Lottery Contests Jingfeng Luy

Zhewei Wangz

April, 2015

Abstract In this paper, we identify a set of axioms that is necessary and su¢ cient for axiomatizing the reverse nested lottery contest proposed by Fu, Lu and Wang (2014), which is the “mirror image” of the conventional nested lottery contest of Clark and Riis (1996). This paper thus provides an axiomatic underpinning for the reverse model, which further reveals the connections and di¤erences between the two (conventional and reverse) multi-prize contest models. JEL Nos: C72; D72; D74. Keywords: Contest Elimination Function (CEF); Contest Success Function (CSF); Conventional Nested Lottery Contest; Reverse Nested Lottery Contest; Axiomatization.

1

Introduction

In the literature on contests, the conventional nested lottery contest model proposed by Clark and Riis (1996) has been widely adopted to study contests with multiple prizes. In a recent paper, Fu, Lu and Wang (2014) propose an alternative multi-prize contest model, i.e., the reverse nested lottery contest, which can be viewed as the “mirror image”of the conventional model of Clark and Riis (1996). While the model of Clark and Riis is de…ned based on the socalled Contest Success Function (CSF), the reverse model of Fu, Lu and Wang relies on the notion of the Contest Elimination Function (CEF), which provides a contestant’s probability of being ranked the lowest among a group of contestants given their e¤ort outlays. Using the We thank Dan Kovenock, Chenzhong Qin, Birendra Rai, Christian Riis, Santiago Sanchez-Pages, Stergios Skaperdas and Alberto Vesperoni for their kind comments and discussions. All remaining errors are ours. y Jingfeng Lu: Department of Economics, National University of Singapore, 10 Kent Ridge Crescent, SINGAPORE, 119260; Tel: (65)65166026, Fax: (65)67752646, Email: [email protected]. z Zhewei Wang: School of Economics, Shandong University, 27 Shanda Nanlu, Jinan, CHINA, 250100; Tel: (86)53188369992, Fax: (86)53188571371, Email: [email protected].

CEF, the reverse model can be conveniently described as determining winners by eliminating contestants sequentially through a hypothetical sequence of lotteries. Lu and Wang (2015) axiomatize the conventional nested lottery contest model by identifying a necessary and su¢ cient set of six axioms that exactly characterize the conventional model. Speci…cally, they …nd that a sixth axiom of “Independence of Irrelevant Ranks” (IIR), together with …ve axioms adapted from Skaperdas (1996),1 is necessary and su¢ cient. In this paper, we address the axiomatization of the reserve nested lottery contest. We …rst establish that the lottery-form CEF adopted by Fu, Lu and Wang (2014) is equivalent to …ve axioms, which are mirror images of the …rst …ve axioms of Skaperdas (1996). We further show that the reverse nested lottery contest is exactly characterized by introducing a sixth axiom, which is a mirror image of Axiom 6 (Independence of Irrelevant Rankings) in Lu and Wang (2015). In other words, the six axioms constitute the necessary and suf…cient conditions for deriving the reverse nested lottery contest. Our paper thus provides an axiomatic underpinning for the lottery-form CEF and reverse nested lottery contests. This axiomatic framework further reveals the connections and di¤erences between the conventional and reverse nest lottery contest models (built on CSF and CEF respectively) and provides guidance for their applications in di¤erent competitive circumstances. Our study belongs to the contest axiomatization literature. Most works in this literature concern with single prize contests. Skaperdas (1996)’s pioneer work axiomatizes the (singleprize) lottery-form CSF. Clark and Riis (1998) generalize the axiomatic characterization to unfair contests. Münster (2009), Rai and Sarin (2009), Arbatskaya and Mialon (2010) and Cubel and Sanchez-Pages (2015) further accommodate multi-dimensional e¤orts. Blavatskyy (2010) axiomatizes the CSFs when the possibility of a draw is present. Our paper is closely related to Vesperoni (2015) who axiomatizes a new multi-prize contest model by assuming axioms imposed on relative probability, which is the ratio of probability of a ranking and that of its pair-swap ranking.

2

Axiomatizations

2.1

Model settings

Following Lu and Wang (2015), let N = f1; 2; :::; ng denote the contestants/players and YN = (y1 ; y2 ; :::; yn ) denote their one-shot e¤ort entries, where player i’s e¤ort is yi 2 R+ , 8i 2 N. Without loss of generality, there are n (weakly) decreasing non-negative prizes denoted by V = (v1 ; : : : ; vn ), with v1 ::: vn 0. We focus on situations where 1

Namely, axioms: (A1) imperfect discrimination; (A2) monotonicity; (A3) anonymity; (A4) sub-contest consistency; and (A5) independence from irrelevant contestants (IIC).

2

each player wins and only wins one prize. As the ranking of the n prizes is …xed, every possible prize allocation outcome can be uniquely determined by a (strict) complete ranking of players.2 A multi-prize contest is essentially characterized by the probabilities of all possible prize allocation outcomes for each YN . Equivalently, a multi-prize contest can be fully characterized by the probabilities of all possible (strict) complete ranking outcomes of n players for each YN . Let RN denote the set of all possible (strict) complete ranking outcomes of players N, and rN 2 RN . Note there are n! elements in set RN , and a representative element can be written as a vector rN = (ik )nk=1 where ik denotes the player who is ranked at the k-th highest position. We use r(i; rN ) to denote the rank of player i 2 N in ranking outcome rN , i.e., r(ik ; rN ) = k. A multi-prize contest is equivalent to its associated Contest Ranking Function (CRF), which is de…ned as below. De…nition 1 A contest ranking function (CRF) p(rN ; YN ) derived from a multi-prize contest speci…es the probability that each possible ranking outcome rN 2 RN occurs for any feasible e¤ort entries YN 2 [R+ ]n . Let M = fj1 ; j2 ; :::; jjMj g N denote a subgroup of jMj di¤erent players in N with 2 jMj n. Let YM = (yj1 ; yj2 ; :::; yjjMj ) denote their e¤ort entries. We de…ne the CRFderived probability that a player j 2 M is ranked the lowest within M as follows. De…nition 2 Let

2.2

j M (YN )

=

P

8rN 2RN s.t.: r(j;rN ) r(l;rN );8l2M

p(rN ; YN ); 8j 2 M

N.

Axiomatization of the reverse nested lottery contests

De…ne N0 = fi 2 Njyi = 0g and M0 = fi 2 Mjyi = 0g, which are two sets of players in N and M who exert zero e¤ort, respectively. The following …ve axioms R1–R5 are imposed on iM (YN ), 8i 2 M N. They are mirror images of Axioms 1–5 of Lu and Wang (2015), which are adapted from Skaperdas (1996). P Axiom R1 Imperfect Discrimination: (i) i2N iN (YN ) = 1 with iN (YN ) 0, 8i; 8YN . (ii) 1 i i i N (YN ) > 0, 8i 2 N if N0 = ;. (iii) N (YN ) = 0, 8i 2 NnN0 and N (YN ) = jN0 j , 8i 2 N0 , if N0 6= ;. Axiom R2 Monotonicity: iN (YN ) weakly decreases with yi and weakly increases with yj , 8i; j 2 N, j 6= i. In particular, iN (YN ) strictly decreases with yi when iN (YN ) > 0 and strictly increases with yj when iN (YN ) > 0; jN (YN ) > 0, 8i; j 2 N, j 6= i. Axiom R3 Anonymity: Let 2

be any permutation of N, i.e., a bijection

: N ! N. De…ne

Here “complete” means that all players are ranked, and “strict” means that players are ranked without

ties.

3

YN = (yi ) where agent i’s e¤ort is yi = y 8 ; 8YN .

1 (i)

; 8i 2 N. Then

(i) N (YN )

=

i N (YN ); 8i

2 N,

Axiom R1 states that the candidate model must satisfy the conditions for a probability distribution function.3 Axiom R2 says that a player’s probability of being ranked the lowest is decreasing in his own e¤ort and increasing in every other player’s e¤ort. Axiom R3 is an anonymity property, which implies that players’probabilities of being ranked at the bottom are equal if they were to exert identical e¤orts.4 Axiom R4 Sub-contest Consistency: 8M i M (YN )

=P

N, for all player i 2 M,

i N (YN ) j j2M N (YN )

if

P

j2M

j N (YN )

> 0:

(1)

Axiom R5 Independence of Irrelevant Contestants (IIC): 8i 2 M N, iM (YN ) = iM (YM ) where YM = (yj )j2M , i.e., player i’s probability of ranking the lowest among M is independent of e¤orts of the players not included in M. Axiom R4 requires that in the original contest involved all players in N, the probabilities of being ranked the lowest among a subgroup of players are qualitatively similar to those among all n players. Axiom R5 requires that in the original contest involved all players in N, a player’s probability of ranking the lowest among a group of players is independent of e¤orts of players outside this group. The following proposition shows that the lottery-form CEF of Fu, Lu and Wang (2014) is exactly characterized by Axioms R1-R5. Proposition 1 Axioms R1–R5 are necessary and su¢ cient for deriving iM (YN ), which takes exactly the form of the lottery-form Contest Elimination Function (CEF) as de…ned in Fu, Lu and Wang (2014):5 P (i) iM (YN ) = iM (YM ) = f (yi ) 1 =( j2M f (yj ) 1 ), 8i 2 M N, where f ( ) is a nonnegative increasing function with f (0) = 0, if jM0 j = 0; (ii) iM (YN ) = iM (YM ) = 1=jM0 j, 8i 2 M0 and jM (YN ) = jM (YM ) = 0, 8j 2 MnM0 , if jM0 j 1. Proof. It is straightforward to show that probabilities piM (YN ) which satisfy the conditions of Proposition 1 must satisfy Axioms R1–R5. We next show that Axioms R1–R5 are su¢ cient. 3

To address the zero-e¤ort case, more speci…cally, Axiom R1 assumes that: A zero e¤ort would generate a strictly positive chance of being ranked the lowest. A strictly positive e¤ort would generate a zero chance of being ranked at the bottom if there exists some player(s) exerting zero e¤ort. Each of the players exerting zero e¤orts gets an equal chance of being ranked at the bottom. 4 Note that this result implies that we have iN (0) = 1=jNj = 1=n, which is consistent with Axiom R1. 5 It was initially called the “ratio-form CEF” in Section 2.1 of Fu, Lu and Wang (2014).

4

i N (YN )

We …rst assume jN0 j = 0. It is clear that Axioms R4 and R5, we have 8i; j 2 M N, i N (YN ) j N (YN )

i M (YM ) j M (YM )

=

> 0; 8i 2 N by Axiom R1 (ii). By

i fi;jg (yi ; yj ) : j (y ; y ) i j fi;jg

=

(2)

P N, As Axioms R4 and R5 also imply iM (YM ) = iN (YN )=( j2M jN (YN )), 8i 2 M i i N; 8YN , the Anonymity property of N (YN ) (Axiom R3) extends to M (YM ); 8i 2 M 8YN , which is described as follows. Let M be any permutation of M, i.e., a bijection M : M ! M. De…ne YM = (yi ; i 2 M) where agent i’s e¤ort is y (i) = yi , 8i 2 M. Then (i) M (YM )

= iM (YM ); 8i 2 M; 8 M ; 8YM , where function iM ( ) only depends on M through jMj but not the identity of M. Thus, there exists a function jMj ( ) such that iM (YM ) = N; 8YM . Moreover, by jMj (yi ; YMnfig ) where YMnfig = (y1 ; :::; yi 1 ; yi+1 ; :::; yjMj ); 8i 2 M P Axioms R1 and R4, i2M jMj (yi ; YMnfig ) = 1. Therefore, we have 8i; k 2 M N, 1 =

i M (YM ) [P ][ j j6=i;j2M M (YM )

= [ = [

1 1

k M (YM ) ][ i M (YM )

jMj (yi ; YMnfig )

][

jMj (yi ; YMnfig )

][

jMj (yi ; YMnfig ) jMj (yi ; YMnfig )

P

j j6=i;j2M M (YM ) ] k M (YM )

jfi;kgj (yk ; yi )

][

P

jfi;kgj (yi ; yk ) j6=i;j2M

1

2 (yi ; yk ) 2 (yi ; yk )

Let f (yi ; yk ) =

1

][1 +

jfj;kgj (yj ; yk )

jfj;kgj (yk ; yj )

P

2 (yj ; yk )

j6=i;j6=k;j2M

2 (yi ; yk ) 2 (yi ; yk )

]

(1

2 (yj ; yk ))

]:

(3)

(4)

;

8i; k 2 N. Substituting (4) to (3), we have 1=[

1

jMj (yi ; YMnfig )

jMj (yi ; YMnfig )

P

][f (yi ; yk )][1 +

j6=i;j6=k;j2M

which implies M (YM )

=

jMj (yi ; YMnfig )

=

Let M = N in (5), we have i N (YN )

=

f (yj ; yk ) 1 ];

1+

1+

f (yi ; yk ) 1 j6=k;j2M f (yj ; yk )

P

f (yi ; yk ) 1 j2N;j6=k f (yj ; yk )

P

1

:

Using (6) and Axiom R4, it is straightforward to derive that: 8M is an arbitrary player in N and jMj 2, i M (YM )

=P

f (yi ; yk ) 1 j2M f (yj ; yk ) 5

1

:

1

:

(5)

(6) Nnfkg where player k

(7)

By Axiom R5 (IIC), (7) is independent of yk . De…ne f (x) = f (x; 1), then we have i M (YM )

f (yi ; yk ) 1 j2M f (yj ; yk )

=P

1

f (yi ; 1) 1 j2M f (yj ; 1)

=P

As k is an arbitrary player in N, we have 8M i M (YM )

=P

N where 2

f (yi ) 1 j2M f (yj )

1

1

=P

f (yi ) 1 j2M f (yj )

jMj

.

n

1

.

1, (8)

Combining (2) and (8), we have 8i; j 2 N;

i N (YN ) j N (YN )

Using (9) and

P

j2N

j N (YN )

=

f (yi ) f (yj )

1 1

(9)

:

= 1 (Axiom R1), we have

i N (YN )

=P

i N (YN ) j j2N N (YN )

=P

f (yi ) 1 j2N f (yj )

1

,

P which, by Axioms R4, further implies iM (YN ) = iM (YM ) = f (yi ) 1 =[ j2M f (yj ) 1 ]. So Proposition 1 holds when jN0 j = 0. We now assume jN0 j 1. Consider two subcases: Case I with jN0 j = n and Case II with 1 jN0 j n 1. First look at Case I with jN0 j = n, i.e., YN = 0. Axioms R1 (iii) implies that iN (YN ) = 1=n. Take any M N and jMj 2. Axiom R4 gives i i M (YN ) = M (YM ) = 1=jMj = 1=jM0 j. Thus Proposition 1 holds. We now look at Case II with 1 jN0 j n 1. Take any M N and jMj 2. If jM0 j = 0, de…ne YN = (yi ) = (YM ; Y~NnM ) where Y~NnM is taken such that y~j > 0; 8j 2 NnM. Note yi > 0; 8i. By a similar procedure for the case with jN0 j = 0, we have iN (YN ) = P P f (yi ) 1 =( j2N f (yj ) 1 ). Axioms R4 and R5 imply iM (YN ) = f (yi ) 1 = j2M f (yj ) 1 = P f (yi ) 1 =( j2M f (yj ) 1 ) = iM (YN ). Thus Proposition 1 holds. We now assume jM0 j 1 in Case II (where 1 jN0 j n 1). Axioms R1 (iii) means that iN (YN ) = 1=jN0 j, 8i 2 N0 and iN (YN ) = 0, 8i 2 NnN0 . So by Axiom R4, we have j j i i M (YN ) = M (YM ) = 1=jM0 j; 8i 2 M0 and M (YN ) = M (YM ) = 0; 8j 2 MnM0 . Thus Proposition 1 holds. Additional axiom(s) are needed in order to pin down the CRF and the following sixth “Independence from Irrelevant Ranks”(IIR) axiom serves this purpose in the sense that the whole set of six axioms constitutes necessary and su¢ cient conditions for axiomatizing the reverse nested lottery contests. The following IIR requires that the ranking outcomes of any group of lower ranked players have no impact on the ranking outcomes of the rest. We …rst introduce some notations before writing down the IIR Axiom. Suppose players in group L are ranked as the jLj lowest among all players in N, where 0 jLj n 2. Let 6

rL denote a complete ranking of players in L given they are ranked as the jLj lowest in the original n-player contest, and let RL denote the set of all these possible complete rankings rL of L. Let M NnL and iM (YN jrL ) denote the probability of player i 2 M being ranked the lowest within M conditional on rL (for given e¤ort outlays YN ). Axiom R6 Independence of Irrelevant Rankings (IIR): Given players L have been ranked the jLj lowest among N, 8i 2 M NnL, his probability of being ranked the lowest within M is independent of the ranking rL of L, i.e., iM (YN jrL ) = iM (YN ). The IIR property (Axiom R6) indicates that a player’s losing probability among M does not depend on the ranking outcomes among L NnM given that players in L have been ranked the lowest jLj positions. In the reverse nested lottery contest model of Fu, Lu and Wang (2014), players N simultaneously submit their one-shot e¤ort entries YN to compete for n prizes denoted by V. Each contestant is eligible for one prize. The recipients of prizes are determined by selecting losers through a sequence of independent “lotteries” among all remaining eligible candidates. The draws for lower prizes are conducted earlier.6 Given players’one-shot e¤ort entries YN , a single prize recipient is drawn in each lottery draw. The recipient of a prize is immediately removed from the pools of players who are eligible for future draws. This procedure is repeated until all the prizes are given away. Let k denote the set of all eligible players for the k-th draw for the k-th prize vk ; k = 1; 2; :::; n. The probability that a player P i 2 k receives prize vk is given by a lottery-form CEF: i k (YN ) = f (yi ) 1 =( j2 k f (yj ) 1 ) if f (yj ) > 0, 8j 2 k .7 The CRF derived from the reverse nested lottery contest is given by the following Lemma. Lemma 1 Consider a reverse nested lottery contest of Fu, Lu and Wang (2014) with players N and impact function f (yi ), i 2 N, where f ( ) is increasing in its argument with f (0) = 0. The derived CRF that generates a probability of an arbitrary (strict) complete ranking of players rN = (ik )nk=1 2 RN for given e¤ort outlays YN is given by (i) p(rN ; YN ) = (ii) p(rN ; YN ) = if jN0 j

f (yik ) 1 n P k=1 k 1 k0 =1 f (yik0 )

if jN0 j = 0;

f (yik ) 1 n jN0 j Pk k=1 1 k0 =1 f (yik0 )

1 and yik = 0; 8k

(iii) p(rN ; YN ) = 0 if jN0 j

n

1 and 9k

6

!

jN0 j 1 h=1

h

jN0 j + 1;

n

jN0 j + 1; yik > 0:

(10)

This di¤ers from the conventional nested lottery contest model of Clark and Riis (1996a) where the draws for higher prizes are conducted earlier. 7 When 9j 2 k , yj = 0, the probability that player i 2 k wins prize vk is 1=[#(jjf (yj ) = 0; j 2 k )], where #(jjf (yj ) = 0; j 2 k ) is the count of zero e¤orts among k .

7

The following theorem presents our main result. Theorem 1 Axioms R1–R6 are satis…ed if and only if the derived CRF is given by (10). Proof. [IF Part:] We …rst show that the IIC property (Axiom R5) is satis…ed by the CRF described in Lemma 1. Take any M such that 2 jMj n 1. When jM0 j 1, then jN0 j 1, we show that Axiom R5 holds from Lemma 1 and De…nition 2 in the following. When jM0 j 1, if i 2 MnM0 , we derive iM (YN ) = iM (YM ) = 0 from Lemma 1 (iii) and De…nition 2; if i 2 M0 , by Lemma 1 (ii) and De…nition 2, we derive iM (YN ) = iM (YM ) = P 1=jM0 j. When jM0 j = 0, we next show that iM (YN ) = iM (YM ) = f (yi ) 1 =( j2M f (yj ) 1 ). Consider the following random performance ranking model of Fu, Lu and Wang (2014) with multiplicative noise: xi = f (yi ) "i ; 8i 2 N, where xi is the perceivable output of player i and yi is his e¤ort, the noise term "i follows a Weibull (minimum) distribution. Noises "i are independent across i. Prizes V are allocated according to the rankings of observed outputs xi ; i 2 N. Each player wins at most one prize. Higher outputs win higher prizes. Ties are broken randomly with equal winning chances. Theorem 2 in Fu, Lu and Wang (2014) implies that the above model generates exactly the same CRF as described by Lemma 1 if N0 = ;.8 Therefore, the above noisy performance ranking model is stochastically equivalent to any model that generates the CRF of Lemma 1. We thus have that iM (YN ) (De…nition 2) derived from the CRF of Lemma 1 must coincide with the probability Pr(xi < xj ; 8j 2 MjYN ). By Theorem 1 of Fu, Lu and Wang (2014), we have P Pr(xi < xj ; 8j 2 MjYN ) = Pr(xi < xj ; 8j 2 MjYM ) = f (yi ) 1 =( j2M f (yj ) 1 ) as f (yj ) > 0, 8j 2 M. We thus have shown that Axiom R5 holds by the CRF described in Lemma 1. Based on the stochastic equivalence pointed out as above, the IIR property (Axiom R6) can be obtained using Theorem 2 of Fu, Lu and Wang (2014). Sub-contest Consistency propP erty (Axiom R4) holds as we have established above that iM (YN ) = f (yi ) 1 =( j2M f (yj ) 1 ), P 8M N. When jN0 j = 0, iN (YN ) = f (yi ) 1 =( j2N f (yj ) 1 ), where f ( ) increases with its argument, and when jN0 j 1, iN (YN ) = 1=jN0 j if i 2 N0 and iN (YN ) = 0 if i 2 NnN0 , Imperfect Discrimination property (Axiom R1), Monotonicity property (Axiom R2) and Anonymity property (Axiom R3) must therefore hold. [ONLY IF Part:] Next, we are going to show that Axioms R1–R6 must render the CRF of Lemma 1. In the following analysis we focus on the case where jN0 j = 0, while the other cases with jN0 j 1 can be taken care of following a similar procedure. By Proposition 1 (i), we derive that for the CRF satisfying Axioms R1–R5, the probability of player i being ranked the lowest in an n-player contest must be in a form iN (YN ) = 8

It is straightforward to extend the result to the cases where N0 6= ; by allowing e¤orts of some players are zero, the proof is omitted to save space.

8

P f (yi ) 1 =( j2N f (yj ) 1 ). Substituting it into (1) in Axiom R4 yields ! ! 1 1 X f (y ) f (y ) f (yi ) 1 i j i P P P = = M (YN ) = 1 1 g2N f (yg ) g2N f (yg ) j2M f (yj ) j2M

1

:

(11)

By Axioms R5 and R6, the probability of a complete ranking rN = (ik )nk=1 is p(rN ; YN ) = in 1 i2 i3 in ::: N (YN ) fi2 ;i1 g (YN ). Substituting (11) into the above fi3 ;i2 ;i1 g (YN ) Nnfin g (YN ) P equation, we have p(rN ; YN ) = nk=1 [f (yik ) 1 =( kk0 =1 f (yik0 ) 1 )], which is exactly the form of CRF of Lemma 1 (i).

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