Voluntary Leadership in Teams and the Leadership Paradox

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1 Voluntary Leadership in Teams and the Leadership Paradox Susanne Mayer 1 and Christian Hilbe 2 1 Institute for Social Policy, Department Socioeconomics, Vienna University of Economics and Business, Nordbergstraße 15, 1090 Vienna, Austria 2 Research group for Evolutionary Theory, Max-Planck Institute for Evolutionary Biology, August-Thienemann-Straße 2, Plön, Germany. June 10, 2013 Abstract Recent game theoretic research emphasizes the role of productivity spillovers for leadership in teams of two players: leadership emerges as the first mover s efforts improve the follower s productivity, and thereby both players intrinsic motivation to contribute. Herein, we investigate the role of productivity spillovers in teams of n players. We find a leadership paradox: even if the leader s contributions enhance the productivity of all followers, leadership may not be stable, due to strategic interactions among the followers. However, we also prove that such a leadership paradox does not occur if the group of followers is sufficiently homogeneous, or if the followers contributions are complements. 1 Introduction Many successful real-world team efforts owe to one leader taking the reins, motivating others to oin in and preventing potential beneficiaries from free-riding. A key question thus is how these leaders trigger their peers to have a share in the oint proect and why leadership, unless predetermined exogenously, emerges in the first place. To be stable, leadership needs to be universally accepted; leaders have to prefer to take the lead, and followers need to have an incentive to follow. Such incentives could arise due to information asymmetries (Hermalin, 1998), or behavioral motives such as warm-glow (Romano and Yildirim, 2001) or conformist bias (Huck and Rey-Biel, 2006). Moreover, in the context of international policy in a two-country model, it was argued that sequential move structures are Pareto-efficient if the countries contributions are complements (Kempf and Rota- Graziosi, 2010a). Applied to teamwork, this implies that leadership will emerge if the leader is able to trigger positive productivity spillovers, i.e. if the leader s contributions increase the follower s productivity. Herein, we extend this theory to teams of n players, investigating under which conditions positive productivity spillovers give rise to leadership. 1

2 2 The model We model teamwork as a public goods game between n agents. All agents are endowed with M i units of a numeraire from which they may contribute x i [0, M i ] to the public good. The total amount of the public good produced is G(x), where x =,..., x n ) is the vector of individual contributions and G is the public good s production function. We do not require G(x) to depend on total contributions only, G(x) = g ( n i=1 x i); such a specification would imply that individual contributions are perfect substitutes and that higher contributions of one agent lead naturally to a crowding-out of the other agents contributions (Varian, 1994). Instead, G is only assumed to be twice continuously differentiable and strictly increasing in all arguments, i.e. x i G(x) > 0 for all i {1,..., n} and for all contribution tuples x. We interpret the value of G/x i as agent i s productivity. An agent s productivity may be affected by several factors; in particular, it may depend on the contributions of the other agents. If further contributions of agent increase i s productivity, that is if ( ) G(x) > 0 for all x, (1) x x i then we say that there are positive productivity spillovers between i and, as in the example of the Cobb-Douglas function G(x i, x ) = x 1/2 i x 1/2. If, on the other hand, the cross-partial derivative on the left side of Equ. (1) is negative for all x, then there are negative productivity spillovers, as for example when G(x i, x ) = (x i + x ) 1/2. Each agent benefits from the consumption of the public good and from the remaining numeraire, u i (x) = G(x) + v i (M i x i ), where the function v i describes the private utility of the numeraire. The utility functions of all agents as well as the production technology of the public good are common knowledge. To investigate the stability of leadership, we compare the Nash equilibrium (x N 1, xn 2,..., xn n ) of the game where all players move simultaneously with the subgame perfect equilibrium (x L 1, xf 2,..., xf n ) of the sequential game with the exogenously determined leader, player 1, moving first and with all other players moving second. To this end, let us assume that the production technology G(x) is concave in each argument, and that v i fulfills the Inada conditions, such that the respective equilibria for the simultaneous game and for the sequential game are unique and in the interior of the strategy space. 3 Leadership in homogeneous groups Let us first analyze the simplest possible extension of the two-player case, assuming that all followers are identical. This assumption implies that all followers have the same endowment, and they derive the same utility from the consumption of the private numeraire, i.e. M i = M and v i = v for all i {2,..., n}. Moreover, G(x) is independent of the followers labels, that is G, x 2,..., x n ) = G, x i2,..., x in ) for any permutation {i 2,..., i n } of the set {2,..., n}. Proposition 1 Let f (x ) be player s best response function with respect to the co-players contribu- 2

3 tions x =,..., x 1, x +1,..., x n ). (i) If there are positive productivity spillovers between the leader and the followers, then the followers contributions are positively related to the leader s contributions, x 1 f (x ) > 0. On the other hand, if there are negative productivity spillovers, then x 1 f (x ) < 0. (ii) As a consequence, positive productivity spillovers between leader and followers result in higher individual contributions in the sequential game, that is (x L 1, xf 2,..., xf n ) > (x N 1, xn 2,... xn n ). On the other hand, for negative productivity spillovers the leader s contributions in the sequential game decrease, x L 1 < xn 1, whereas the contributions of the followers increase, x F > xn for all followers. (iii) For positive productivity spillovers between leader and followers, all players prefer the sequential move structure with player 1 as the leader to a simultaneous game. On the other hand, for negative productivity spillovers the potential followers prefer the simultaneous game to the sequential game. For negative productivity spillovers, this result essentially reproduces Varian (1994): if contributions are substitutes, then the sequential game results in a reduction of the leader s efforts, thereby decreasing the followers welfare. In such a case, leadership may not emerge, as the followers have no incentive to accept a sequential move structure. In contrast, if the leader triggers positive productivity spillovers, then all agents extend their efforts in the sequential game, and the presence of a leader can successfully raise the level of overall contributions. Note that the assumption of identical followers is trivially met if n = 2; it is reassuring that our results for this special case are in agreement with the two-country model of Kempf and Rota-Graziosi (2010a). Interestingly, the qualitative results in Proposition 1 only depend on the sign of the productivity spillovers between the leader and the followers. Contrariwise, the sign of the productivity spillovers between the followers themselves are irrelevant. This result is a consequence of our assumption of homogenous groups, as we shall see in the next section. 4 Leadership in heterogeneous groups In many realistic cases, it will be more appropriate to assume that the group of followers is heterogeneous. Clearly, such games in heterogeneous groups offer a considerably richer complexity of dynamic interactions; the leader does not only need to consider how her contributions affect the productivity of the followers, but she must also take into account the potential interdependencies among the followers contributions. In fact, increased efforts of one follower may be overcompensated by reduced contributions of several other followers if the followers contributions are substitutes to each other. To elucidate this point, let us assume that each team member is endowed with one monetary unit, that the utility of private consumption of the numeraire is given by v i (M i x i ) = 1 x 2 i, and that the team produces a public good according to the polynomial production function G(x) = b i x i + s i x i x. (2) 1 i n 1 i< n 3

4 The parameters b i > 0 reflect the baseline productivity of each player, whereas the coefficients s i correspond to the players productivity spillovers. It is straightforward to calculate the players best response functions, f (x ) = 1 b + s i x i. (3) 2 i Thus, in the case of positive productivity spillovers between the leader and all followers (s 1 > 0 for all > 1), an increase of the leader s efforts leads, ceteris paribus, to an increase of all followers contributions. As a mutual increase of contributions is beneficial for the whole group, one may therefore expect that leadership again arises endogenously. However, depending on the productivity spillovers between the followers (i.e., depending on the values of s i for 2 i < < n), an initial increase of all contributions may motivate some of the players to actually reduce their efforts in the eventual equilibrium. In extreme, this can result in a paradox: although further contributions of the leader would enhance the productivity of all followers, the leader may actually cut down her contributions in the subgame perfect equilibrium. This paradox is illustrated in Figure 1, showing the best response functions of the two followers, player 2 and player 3, given that player 1 chooses the Nash-contribution x N 1 (solid lines), or the subgame perfect contribution xl 1 (dashed lines). Although further contributions would increase the productivity of both followers, the leader decreases her contributions in subgame perfect equilibrium, such that both followers best response functions shift to the left. This results in lower contributions of player 2, and in higher contributions of player 3. Since both other players reduce their contributions, player 3 has an incentive to oppose player 1 being a leader. The next proposition shows that negative productivity spillovers between the followers are indeed necessary for such a leadership paradox to arise. Proposition 2 Suppose the public good s production function admits positive productivity spillovers only, i.e. inequality (1) holds for all i, {1,..., n}. Then (x L 1, xf 2,..., xf n ) > (x N 1, xn 2,..., xn n ), and all players prefer the sequential move structure to the simultaneous game. 5 Discussion A recent paper by Kempf and Rota-Graziosi (2010a) emphasizes the role of positive productivity spillovers for voluntary leadership in a two-country model. In our article, we interpret their model as a model of teamwork and extend it to teams of n players. For homogenous groups, we confirm that positive productivity spillovers between the followers and their potential leader give rise to the endogenous emergence of leadership. For heterogeneous teams, however, we find an interesting counter-example: even if the leader s further contributions would enhance the productivity of all co-players, it may be rational for a first-mover to actually decrease her contributions. However, we also show that such a counter-example can only occur if the followers contributions are substitutes, such that 4

5 Contributions of player 3 x 3 F x 3 N Subgame perfect equilibrium Nash equilibrium f 2 N,x3 ) f 3 N,x2 ) f 2 L,x3 ) f 3 L,x2 ) x 2 F x 2 N Contributions of player 2 Figure 1: An illustration of the leadership paradox. Parameters in Equ. (2) are given by b 1 = 1, b 2 = 1/3, b 3 = 2/3, and s 12 = 1/2, s 13 = 1/10, s 23 = 3/2. For these parameters, the Nash equilibrium is (x N 1, x N 2, x N 3 ) (0.533, 0.068, 0.309), whereas the subgame perfect equilibrium is (x L 1, x F 2, x F 3 ) (0.418, 0.013, 0.345). higher contributions of one follower may lead to a crowding-out of the contributions of other followers. In such a case, at least one follower will prefer a simultaneous game to the sequential game, making leadership inherently unstable. Although we have framed our analysis as a model of teamwork, the leadership paradox may be interpreted more generally. In fact, similar problems may arise between interdependent public institutions such as governments (Kempf and Rota-Graziosi, 2010a), or for sequential donations to charities (Romano and Yildirim, 2001). Also in these applications, a ceteris paribus positive effect of the leader on the followers cooperation may be not sufficient for leadership to arise. We believe that our model allows various avenues for future research. First, we have distinguished between two possible move orders only, comparing a simultaneous game with a sequential game with player 1 moving first. However, in many teams there will be a variety of potential leaders, which can result in a coordination problem as players may disagree on the optimal move structure (Kempf and Rota-Graziosi, 2010a, 2010b). This is of particular importance if agents are asymmetrically informed about the public good s production technology, such that they might benefit from selecting the best informed agent as their leader. Finally, the question of leader identity could also be taken as a starting point for future experiments, challenging productivity spillovers as a managerially intuitive rationale for leadership in teams. Appendix: Proofs of the Propositions Proof of Proposition 1. (i) Let (y, x ) denote the contribution vector,..., x 1, y, x +1,..., x n ). Then 5

6 player s best response function f (x ) is implicitly given by h(y) = x G(y, x ) v (M y) = 0. (A.4) Since h (y) 0, we may differentiate Eq. (A.4) with respect to x 1, yielding the following expression for the partial derivative x 1 f (x ) = 2 x 1 x G(f (x ), x ) 2 G(f x 2 (x ), x ) + v (M f (x )). (A.5) As the denominator of x 1 f (x ) is negative, the result follows. (ii) We analyze the sequential game by backward induction. For a given contribution x 1 of the leader, the game among the followers is symmetric and hence has a symmetric equilibrium. Let g ) denote a follower s equilibrium contribution, and let G, g )) = G, g ),..., g )) denote the respective amount of public good produced. As before, the followers common reaction function g ) is implicitly given by x G, g )) v (M g )) = 0 for any {2,..., n}. Differentiating this equation with respect to x 1 yields g ) = 2 x 1 x G, g )) (n 1) 2 G(x x 2 1, g )) + v (M g )). (A.6) Thus g ) is monotonically increasing for positive productivity spillovers, and monotonically decreasing for negative productivity spillovers. Given the reaction of the followers, player 1 aims to maximize u 1, g )) = G, g )) + v 1 (M 1 x 1 ); differentiating this function around the Nash equilibrium yields x 1 u 1 (x N 1,..., x N n ) = x 1 u 1 (x N 1, g(x N 1 )) = x 1 G(x N 1, g(x N 1 )) + (n 1)g (x N 1 ) x G(x N 1, g(x N 1 )) v 1(M 1 x N 1 ) = (n 1)g (x N 1 ) x G(x N 1, g(x N 1 )) (A.7) For positive productivity spillovers, g (x ) > 0, and thus it is beneficial for player 1 to increase her contributions, implying that x L 1 > xn 1, and xf = g(x L 1 ) > g(xn 1 ) = xn for all followers {2,..., n}. For negative productivity spillovers, x L 1 < xn 1 and x F > xn follow similarly. (iii) By playing x L 1 = xn 1, the leader can always guarantee herself the payoff of the simultaneous game, and hence u L 1 = u 1(x L 1, g(xl 1 )) u 1(x N 1, g(xn 1 )) = un 1. For a follower, positive spillovers imply x L 1 > xn 1 and hence un = u (x N 1, g(xn 1 )) < u (x L 1, g(xn 1 )) u (x L 1, g(xl 1 )) = uf. Proof of Proposition 2. Let us denote by g : R R n 1 denote the function that assigns to each contribution x 1 of player 1 the corresponding equilibrium contribution for 6

7 the game among the followers in the second stage. Due to our assumptions, the function g is differentiable; similar to the proof of the previous theorem, it suffices to show that g is monotonically increasing. For this reason, assume that x 1 < x 1. Consider the sequence x (k) = (x k 1,..., xk n) that is recursively defined by x (0) = ( x 1, g )) and x (k+1) = ( x 1, f 2 (x (k) 2 ),..., f n(x (k) n )), where f denotes player s best response function. Since x 1 > x 1, and since the best response functions are monotonically increasing in each argument (due to the assumption of positive productivity spillovers between all players), the sequence x (k) is monotonically increasing. As it is bounded by (M 1,..., M n ), it follows that it converges to some ˆx. For each {2,..., n} we have ˆx = lim k x (k) lim k f (x (k 1) ) = f (lim k x (k 1) = ) = f (ˆx ). Thus, each follower plays a best response and (ˆx 2,..., ˆx n ) = g( x 1 ). As the sequence x (k) was monotonically increasing, it follows that g ) < g( x 1 ). Acknowledgements The authors gratefully acknowledge the positive productivity spillovers from participants at the International Meeting on Experimental and Behavioral Economics in Castellón and at the Economic Science Association conference in New York as well as Maria Abou Chakra, Julián García, Ulrich Berger, August Österle, Roberto Serrano and Hubert Kempf. References References Hermalin, B. E., Toward an economic theory of leadership: Leading by example. Am. Econ. Rev. 88 (5), Huck, S., Rey-Biel, P., Endogenous leadership in teams. J. Inst. Theor. Econ. 162, Kempf, H., Rota-Graziosi, G., 2010a. Leadership in public good provision: a timing game perspective. J. Pub. Econ. Theor. 12, Kempf, H., Rota-Graziosi, G., 2010b. Endogenizing leadership in tax competition. J. Public Econ. 94, Romano, R., Yildirim, H., Why charities announce donations: A positive perspective. J. Public Econ. 81, Varian, H. R., Sequential contributions to public goods. J. Public Econ. 53,