FREE RIDERS DO NOT LIKE UNCERTAINTY *

Transcription

1 Universität des Saarlandes University of Saarland Volkswirtschaftliche Reihe Economic Series FREE RIDERS DO NOT LIKE UNCERTAINTY * Jürgen EICHBERGER AND David KELSEY Economic Series No October 1999 University of Saarland Department of Economics (FB 2) P.O. Box D SAARBRUECKEN, GERMANY Phone: , Fax: j.eichberger@with.uni-sb.de

2 Abstract We examine the effect of Knightian uncertainty in a simple model of public good provision. We find that uncertainty may reduce free-riding if utility is concave or there are decreasing returns to scale in the production of public goods. Comparative statics analysis shows that increases in uncertainty will increase donations. These effects may be reversed if there are sufficiently strong increasing returns to scale. We show that these results generalise and uncertainty tends to increase/decrease equilibrium strategies in games with strategic complements/substitutes and positive externalities. These effects are reversed in games with negative externalities. JEL Codes: D81, H41. Keywords: Public Goods, Strategic Complements, Knightian Uncertainty, Choquet Inte-gral, Free Rider. * Research supported by ESRCsenior research fellowship, award no. H , ESRCgrant no. R and a grant from the School of Social Science at The University of Birmingham. For comments and discussion we would like to thank Richard Cornes, Max Fry, Simon Gächter, Simon Grant, Stephen King, Frank Milne, Sujoy Mukerji, Shasikanta Nandeibam, Yew- Kwang Ng, Thomas Renstrom, Martin Sefton, Peter Wakker and participants in seminars at the Universities of Bielefeld, Birmingham, Essex, Hull, Humboldt (Berlin), Kiel, Mannheim, Nottingham, Penn. State, St. Andrews, Saarlandes, York, Birkbeck College, London, the Econometric society meetings in Iowa City 1996, the Warwick summer workshop 1997 and the European Economic Association Meetings in Toulouse 1997, the referee and associate editor of this journal.

3 1. INTRODUCTION 1.1 Background Conventional models of public goods imply that voluntary provision is virtually impossible in large societies, due to free-rider problems. Despite this, in practice some public goods are privately provided. For instance, in the UK the National Trust (a private charity) preserves buildings and landscapes and receives a large proportion of its funds from voluntary donations. In the USA, many non-commercial radio and television stations are similarly funded. (Other researchers have also noted that experience does not appear to confirm the theory of free-riding, see for instance Isaac, Walker and Thomas [30], p. 113.) Experimental research has similarly found significantly less free riding than is predicted. It has been usual to explain this by postulating that subjects have altruistic preferences. (See, for instance, Sefton and Steinberg [48] or Andreoni [2].) In the present paper we advance an alternative explanation based on Knightian uncertainty. If public goods are financed by voluntary donations, individuals are likely to be uncertain about the contributions of others. Austen-Smith [5] modelled this as risk (additive uncertainty) and argued that risk-aversion (i.e. concave utility) would increase contributions. The model in [5] was partial equilibrium. Sandler, Sterbenz and Posnett [45] and Gradstein, Nitzan and Slutsky [27] showed that once general equilibrium effects were taken into account, risk-aversion was not sufficient to guarantee an increase in contributions. Public good provision would only increase if a certain restriction on the third derivatives of utility, was satisfied. It does not seem unfair to conclude, that there is no general presumption that risk-aversion will increase voluntary contributions to a public good. In the present paper we show that concavity of the utility function will indeed cause contributions to increase, if doubts about the behaviour of others are modelled as Knightian uncertainty rather than risk. 1.2 Knightian Uncertainty Knightian uncertainty refers to situations where it is difficult to assign precise probabilities. It has 2

4 often been argued that this kind of uncertainty is important for economics. Henceforth we shall abbreviate Knightian uncertainty to uncertainty. We shall use the term risk to refer to situations with known probabilities. In this paper we shall use a model of Knightian uncertainty due to Schmeidler [47], which represents individuals beliefs by capacities (non-additive subjective probabilities). He axiomatises preferences that can be represented by maximising the expected value of utility with respect to a capacity. (The expectation is expressed as a Choquet integral, [11].) This theory will henceforth be referred to as Choquet Expected Utility (CEU). Under some plausible assumptions, this gives rise to a form of preference which over-weights the worse outcomes of any given option. In this paper we investigate the influence of uncertainty on the provision of public goods. In our model each individual is uncertain about the contributions of others. Apart from this, the model is deterministic. We believe that it is possible that there may be Knightian uncertainty, where public goods are voluntarily provided. One reason is that economic models are an imperfect reflection of reality. A motivation for Knightian uncertainty is that it models how decision-makers may behave to protect themselves against errors arising from imperfections in the model. This argument is explained in more detail in Mukerji [42]. We find that uncertainty can reduce free-riding by increasing the perceived marginal benefit of contributions, moreover this effect can be quite large. However, even in the presence of uncertainty, the outcome will typically not be Pareto optimal since, each individual still fails to take into account benefits going to others. The sub-optimality of conventional equilibrium arose from this failure. It is only by accident that the perceived increase in marginal benefit due to uncertainty will offset this effect. Hence the equilibrium under uncertainty will not typically be Pareto optimal. Note that this is qualitatively similar to the experimental evidence, which finds sub-optimal provision but significantly less free-riding than would be predicted by conventional theories. 1.3 The Role of Returns to Scale If there are sufficiently strong increasing returns in the production of public goods, then our con- 3

5 clusions are changed. In this case, uncertainty will discourage voluntary contributions. With increasing returns to scale, increases in uncertainty either have no effect or cause a catastrophic collapse in donations (i.e. provision of the public good falls to zero). These results are in contrast to the case of decreasing returns to scale, where we find that uncertainty can increase contributions. In addition, in the concave case, gradual increases in uncertainty caused gradual, rather than catastrophic, increases in donations. However the underlying logic of the two problems is similar. If the production function is sufficiently convex, uncertainty reduces the perceived contributions of others. Since the marginal product of donations is falling, this reduces the marginal benefit of contributing. If the production function is concave, then lower perceived contributions by others increases the anticipated marginal benefit of contributing. Thus voluntary provision is more likely to be successful, the more concave the production and utility functions are. The difference in comparative statics arises from the fact that, with convex functions, optimal points are always at corners, while concave functions usually give rise to interior optima. Another way of viewing this is that, with increasing returns to scale, there are coordination problems in public good provision. A given individual may wish to make a donation if others do so but not otherwise. Hence, there may be multiple equilibria. Uncertainty increases the coordination problem, since individuals are less able to rely on others making contributions. This is because, with increasing marginal product, your donation will be more valuable if others also make donations. In summary, uncertainty may reduce free-riding but will increase coordination problems. Provision of public goods, may either increase or decrease depending upon the production function for public goods. We find it suggestive that, when organisations solicit voluntary contributions, they often emphasise uncertainty concerning future provision and worst case outcomes. (How would you feel if your favourite programme was not broadcast?) This accords well with theoretical models of Knightian uncertainty. Material soliciting donations rarely mentions altruistic motives. 4

6 1.4 Strategic Complements and Substitutes In section 6 we consider a more general game, of which the public goods model is a special case. We investigate how Knightian uncertainty interacts with the properties of strategic substitutes and complements in games. These concepts were introduced by Bulow, Geanakopolos, and Klemperer [9]. In a game with strategic complements (resp. substitutes), if your opponent increases his/her act, the marginal benefit of increasing your own act will increase (resp. decrease). Examples of games with strategic complements and substitutes are respectively Bertrand and Cournot duopolies with linear demand and constant marginal cost. These concepts have proved useful in industrial organisation. They have also been applied in other areas of economics, most noticeably to coordination problems in macroeconomics see for instance Cooper and John [12]. The concept of strategiccomplementarity was generalised to n-player games in Milgrom and Roberts [37],[38] and Milgrom and Shannon [39]. We hope that the analysis in the present paper will facilitate new applications of games with Knightian uncertainty, in particular in the area of industrial organisation. We do not have space in the present paper to consider applications of these results, apart from the public goods model. Further applications, including some in industrial organisation and macroeconomics can be found in a companion paper, [20]. Roughly, we find that Knightian uncertainty has opposite effects in games of strategic complements and substitutes. However this is complicated by the fact that there are possibly multiple equilibria when there are strategic complements. Assume that there are positive spillovers, then uncertainty has the effect of increasing the weight that a player places on the lowest act of his/her opponents. If there are strategic complements (resp. substitutes), this reduces (resp. increases) the marginal benefit of increasing the player s own act and hence the equilibrium values of the strategies. These effects are reversed if there are negative spillovers. If there are positive spillovers, in Nash equilibrium, players will use strategies below the Pareto optimum, hence for small changes increasing uncertainty will move the equilibrium in the direction of the symmetric Pareto if there are strategic substitutes. However uncertainty will move the equilibrium away from the Pareto 5

7 optimum if there are strategic complements. In the case of strategic complements it is possible that there might be multiple symmetric equilibria. These would be Pareto ranked. If there are positive (resp. negative) spillovers then higher (resp. lower) equilibria are Pareto superior. In both cases we find that if there is enough uncertainty only the Pareto inferior equilibrium will arise. As a rough guide, our results may be summarised by saying that uncertainty is helpful in games with strategic substitutes, but tends to have negative effects when there are strategic complements. Organisation of the paper The next section introduces CEU preferences and applies them to games. We present our model of public good provision in section 3. Section 4 characterises the equilibrium and investigates the comparative statics of changing uncertainty. The case of increasing returns to scale is discussed in section 4. Section 5 relates our research to experimental evidence. A more general model is presented in section 6. Related literature is discussed in section 7 and section 8 concludes. The appendix contains the proofs of those results which are not proved in the text. 2. INTRODUCTION TO UNCERTAINTY AND GAMES 2.1 Uncertainty In this paper we model voluntary contributions to the provision of a public good as a non-cooperative game. Traditionally game theory assumes that players have expected utility preferences. We wish to model public good provision in the presence of Knightian uncertainty and hence, assume instead, that the players have CEU preferences. Consider a game =hn;(s i )(u i ):16i 6 ni with finite pure strategy sets S i for each player and payoff functions u i (s i ;s i ). The notation, s i ; indicates a strategy combination for all players except i. The space of all such strategy profiles is denoted by S i. Beliefs about opponents behaviour are represented by capacities. A capacity assigns non-additive weights, which represent beliefs, to subsets of S i. Formally, capacities are defined as follows. 6

8 Definition 2.1 A capacity on S i is a real-valued function º : P (S i )! R (where P (S i ) denotes the set of all subsets of S i ), which satisfies the following properties: 1. A µ B ) º(A) 6 º(B); 2. º(;) =0; º(S i )=1: Below we define a special class of capacities, which will be useful in our analysis. Definition 2.2 A capacity º is called simple if there exists an additive probability ¼ on S i and a real number 2 [0; 1] such that for all events E $ S i ;º(E) = ¼(E). Simple capacities are contractions of additive probabilities. The parameter ; may be interpreted as the individual s confidence in the probability assessment ¼. Definition 2.3 A capacity º is convex (resp. concave) if º(A)+º(B) 6 º(A[B)+º(A\B) (resp. º(A)+º(B) > º(A [ B)+º(A \ B)). If the capacity is convex, CEU preferences tend to overweight the worst outcomes hence they may be termed pessimistic or uncertainty-averse. Concave capacities have the opposite effect and hence may be termed uncertainty-loving. For a convex capacity it is possible that º(A) + º(S i na) < 1, which implies that not all probability mass is allocated to a set and its complement. This expression can be viewed as a measure of the missing probability mass. It is useful to define the following two measures. Definition 2.4 The maximal (resp. minimal ) degree of ambiguity of capacity º is defined by: (º) =1 min (º (A)+º(:A)) (resp: (º) =1 max (º(A)+º(:A)) : AµS AµS The maximal and minimal degrees of ambiguity of the simple capacity º = ¼, defined above, are 1. Note that for a simple capacity the maximal and minimal degrees of ambiguity are equal. We shall use the minimal degree of ambiguity much less frequently. Consequently, if we use the term degree of ambiguity without qualification, we shall mean the maximal degree of ambiguity. These definitions are adapted from Dow and Werlang [17]. They have been justified epistemically by Mukerji [42]. The degrees of ambiguity are measures of the deviation from additivity. For an additive probability they are equal to zero, while for complete uncertainty they 7

9 are equal to one. 1 We can define the support of a capacity as the smallest set of opponents strategies with a complement of capacity zero. For discussion of the definition of support, see Dow and Werlang [18], Haller [29] and Ryan [44]. Definition 2.5 The support of capacity º is a set E µ S i ; such that º (S i ne) = 0 and º (F) > 0, for all F such that S i ne $ F. A possible objection to this definition, is that states outside the support may not be Savagenull. We believe that this argument is not valid because the concept of a Savage-null set was formulated for expected utility and is not appropriate in the present context. In expected utility if a state has positive probability, it always enters into the evaluation of an option, while if it has zero probability it never enters into this evaluation. With CEU preferences, there is in addition a third category of states which may or may not have positive weight depending on how they are ranked. In particular, some states only enter into the evaluation if they yield especially bad outcomes. Dow and Werlang [18] and Lo [35] interpret this third category of states as being infinitely less likely than those which are always given positive weight. The definition of support is quite stringent and only includes states which always enter into the evaluation. In particular, it does not include those states which only count if they yield bad outcomes. Player i has beliefs about his/her opponents behaviour, represented by a capacity º i on S i. The expected payoff from a strategy s i, is expressed as a Choquet integral over S i. Such preferences have been axiomatised by Gilboa [26] and Sarin and Wakker [46]. Schmeidler [47] originally axiomatised CEU, in a slightly different framework. We shall now define the Choquet integral. Notation 2.1 Define u 1 i (s i)=max s i2s i u i (s i ;s i ) and Q 1 i (s i) = argmax s i2s i u i (s i ;s i ). n o For k > 2, define recursively, u k i (s i)=max u i (s i ;s i ) js i 2 S i nq k 1 i (s i ) and Q k i (s i)= 1 Epstein [22] has proposed an alternative measure of uncertainty-aversion. We do not have space in the present paper to present a detailed comparison of the two. We would like to note that, for pure equilibria, which we believe to be the more interesting case, we could obtain similar results using Epstein s measure of uncertainty-aversion. Note that Ghiraradato and Marinacci [25] and Kelsey andnandeibam [32] have proposed formal measuresof ambiguity more in the spirit of the present paper. 8

10 s i 2 S i : u i (s i ;s i ) > u k i (s i) ª.BecauseS i is finite there exists r such that Q r i (s i)=s i. With the convention Q 0 = ;, the Choquet integral can be defined. Definition 2.6 The Choquet integral of u i (s i ;s i ) with respect to capacity º i on S i is: Z rx ³ ³ i V i (s i )= u i (s i ;s i ) dº i = u k i (s i ) hº i Q k i (s i ) º i Q k 1 i (s i ) : 2.2 Games k=1 In this paper we shall consider a symmetric game ; with n players, I = f1; :::; ng : Player i has a finite strategy set which, for convenience, we identify with a subset of the integers, X i = f0; 1;:::;m g ; for i =1;:::;n: All players have the same utility function u (x i ;x i ) ; for i =1;:::;n:In the following we shall only consider symmetric equilibria. We focus on symmetric equilibria since these are standard in the public goods literature, (see for instance Cornes and Sandler [13], p. 161). This enables our results to be compared to those obtained without uncertainty. Asymmetric equilibria are possible with and without uncertainty. The comparison between asymmetric and symmetric equilibria is orthogonal to that between equilibria with and without uncertainty. For this reason, we shall not investigate asymmetric games or equilibria in this paper. We shall use an equilibrium concept based on that of Dow and Werlang [18], which has been extended in [21]. According to this concept players do not randomise but play pure strategies. 2 An equilibrium is an n-tuple of capacities, which describes the beliefs of each player about how his/her opponents will play. In equilibrium we require the support of player i s beliefs to consist of his/her opponents best responses. In the present paper we shall only consider symmetric games and symmetric equilibria. In a symmetric game, all players have the same strategy set and the same utility function, (S = S i ;u= u i ; 1 6 i 6 n). Below we give a formal definition of equilibrium. Definition 2.7 Let be a symmetric game. A capacity º on S i is a symmetric equilibrium of ; if there exists a support, supp º such that for all i :16 i 6 n; supp º µ j6=i R j (º), 2 In [19], we show that individuals with CEU preferences will not have a strict preference for randomisation. This provides a justification for these assumptions. 9

11 where R j (º) = argmax sj2s j R uj (s j ;s j ) dº; is the best response correspondence of player i; given beliefs º. In a symmetric equilibrium, the beliefs of all players are represented by the same capacity º, whose support consists of strategies that are best responses for their opponents. In equilibrium, a player s evaluation of a particular strategy may in part depend on strategies of his/her opponents which do not lie in the support. We interpret these as events a player views as unlikely but which cannot be ruled out. This may reflect some doubts (s)he may have about the rationality of the opponents or whether (s)he correctly understands the structure of the game. Although in the present paper we only consider symmetric equilibria, it is relatively easy to extend this solution concept to the non-symmetric case (see [18], [21], and [36] ). In a related model we prove that all equilibria are symmetric, (see [7] ). Definition 2.8 Let be a symmetric game and let ^º be a symmetric equilibrium of ; if supp ^º contains a single strategy profile we say that it is pure, otherwise we say that it is mixed. Since players choose pure strategies, we are not able to interpret a mixed equilibrium as a randomisation. In a mixed equilibrium some player i say, will have two or more best responses. The support of other players beliefs about i s play will contain some or all of them. Thus an equilibrium, where the support contains multiple strategy profiles, is an equilibrium in beliefs rather than randomisations. We note that even without uncertainty, the concept of equilibrium in beliefs has proved useful, see for instance, Aumann and Brandenburger [4]. If, in addition, it is required that beliefs are additive, then a symmetric equilibrium would be a correlated beliefs equilibrium. If the support consists of a single strategy profile and beliefs are additive, a symmetric equilibrium is a Nash equilibrium. A symmetric equilibrium is a Nash equilibrium, if beliefs are additive and independent, regardless of the size of the support. We have not required that players beliefs be independent, since at present there are unresolved questions concerning the definition of independence for capacities (see Ghirardato [24] ). We shall restrict attention to games which satisfy the following assumption. Definition 2.9 Agame, ; is a symmetric game with aggregate externalities if u (x i ;x i )= 10

12 u (x i ;f(x i )) ; for 1 6 i 6 n,wheref : S i! R is increasing in all arguments. This is a separability assumption. It says that a player only cares about a one dimensional aggregate of his/her opponents strategies. When a player evaluates options by a Choquet integral, the form of the integral will depend on how (s)he ranks possible strategy profiles of his/her opponents. The assumption of aggregate externalities ensures that this ranking remains the same when the individual considers changing his/her play. Note that this assumption does not restrict two player games, hence our analysis can be applied to all such games. The main example of a game with aggregate externalities which we shall consider is that of voluntary contributions to a public good. In this case an individual s utility only depends on his/her own consumption of the private good and the total contribution by others. In particular utility does not depend on how this total is distributed over the other individuals. Cournot oligopoly is also a game with aggregate externalities. Consider a model with n firms 1 6 i 6 n: Let q i and c i (q i ) denote respectively the quantity and the cost of firm i: Suppose that the inverse demand curve is given by P (q) ; where q denotes the total quantity ³ supplied. Then the profits of firm i are given by ¼ i = P q i + P j6=i q j q i c i (q i ) : If we define f (q i )= P j6=i q j; then this satisfies the form of a game of aggregate externalities. There are a number of other examples. For instance if damage to the environment depends on the total polluting activity over all agents (e.g. global warming), then pollution can be modelled as a game with aggregate externalities. Notation 2.2 Since S i is finite, we may enumerate the possible values of f; f 0 <:::<f M : Since f isassumedtobeincreasingf 0 = f (0; :::; 0) and f M = f (m ;:::;m ) : Notation 2.3 Let (x i ;x i )=u(x i ;x i ) u (x i 1;x i ) ; i.e. (x i ;x i ) denotes the marginal bene t to individual i of increasing his/her act from x i 1 to x i when his/her opponents strategy profile is x i : If f (x i )=f r ; we may write (x i ;f r ) for (x i ;x i ) : Note that this is the marginal benefit in the absence of uncertainty. We shall describe later how marginal benefit is modified by uncertainty. 11

13 Notation 2.4 Let be a symmetric game with aggregate externalities we shall use H r (resp. L r ) to denote the event fx i 2 S i : f (x i ) > f r g,(resp.fx i 2 S i : f (x i ) <f r g). Note that H r = :L r : Definition 2.10 A game,, with aggregate externalities is a game of strategic substitutes (resp. complements)if (x i ;f r ) is a strictly decreasing (resp. increasing) function of r. 3 Note that our definition of strategic complements is slightly stronger than the usual one. The usual definition requires that increasing your own strategy increases the marginal benefit to an opponent of increasing his/her strategy. We require that, in addition, there is a monotonic relationship between your strategy and your opponents total benefit. The reason for this being that the marginal benefit under Knightian uncertainty (see Definition 2.13) depends both on the conventional marginal benefit and on how a player ranks his/her opponents actions. Thus, by assuming aggregate externalities we ensure that playing a higher strategy has an unambiguous effect on one s opponents marginal incentives. The following definitions introduce two properties which will be important in determining the effects of Knightian uncertainty. Definition 2.11 Agame,, has positive (resp. negative) spillovers if u (x i ;x i ) is increasing (resp. decreasing) in x j ; for j 6= i: Definition 2.12 A game,, will be said to be concave (resp. convex) if for all i; u (x i ;x i ) is a strictly concave (resp. convex) function of x i. We assume strict concavity because it allows us to state our results in a cleaner way. Similar results could be proved if pay-off functions were weakly concave. The public goods game, discussed in the next section, is a concave game with strategic substitutes and positive spillovers, if there are decreasing returns to scale in the production of public goods. A Cournot game with linear demand and constant marginal cost would be a concave game with strategic substitutes and negative spillovers. 3 We could obtain similar results if the word strictly were omitted from the definition of strategic complements and substitutes. We have not reported these to save space 12

14 2.3 Preliminary Analysis First we define the marginal benefit under uncertainty. Definition 2.13 Marginal Benefit Suppose that player i has beliefs described by capacity º on S i. Define Z Z MB (x i ;º)= u (x i ;x i ) dº(x i ) u (x i 1;x i ) dº(x i ): We interpret MB (x i ;º) as player i s perceived marginal benefit from increasing his/her strategy from x i 1 to x i ; given that (s)he has beliefs represented by capacity º. In general this will be different to the marginal benefit without uncertainty defined in Notation 2.3. Notation 2.5 If 1 6 k 6 m,weshallr (k) use to denote that value of r which satisfies f r(k) = f (k; :::; k) : Lemma 2.1 Let be a game with aggregate externalities, a) assume there are positive spillovers, i) if º is a capacity, M X 1 MB (x i ;º)= (x i ;f M ) º (H M )+ (x i ;f r )[º(H r ) º(H r+1 )] ; ii) if ^º is a symmetric equilibrium with contribution level ^k or contribution levels ^k 1; ^k < m ; then MB (x i ; ^º) = Pr(^k) r=0 (x i;f r )[^º(H r ) ^º(H r+1 )] ; b) assume there are negative spillovers, i) if º is a capacity on S i ; MB (x i ;º)= (x i ;f 0 ) º (L 1 )+ r=0 MX (x i ;f r )[º(L r+1 ) º(L r )] ; r=1 ii) if ^º is a symmetric equilibrium with contribution level ^k or contribution levels ^k; ^k +1, where 0 < ^k <m ; then MB (x i ; ^º) = P M r=r(^k) (x i;f r )[^º(L r+1 ) ^º(L r )] : Remark 2.1 As can be seen, the form of the Choquet integral is different depending on whether there are positive or negative spillovers. This is because the Choquet integral depends on how a given player ranks strategy profiles which might be played by his/her opponents. These rankings are reversed depending on whether there are positive or negative spillovers. Theorem 2.1 (Existence of Equilibrium) Let be a symmetric game of aggregate externalities, then for any degree of ambiguity, there exists a symmetric equilibrium with degree of ambiguity. 13

15 Remark 2.2 Since the proof of the above theorem demonstrates the existence of an equilibrium in simple capacities the term degree of ambiguity in the statement of the theorem can be interpreted as either the maximal or the minimal degree of ambiguity. The following Theorem characterises equilibrium with uncertainty. It is essentially a statement of the usual marginal conditions for equilibrium. The only subtlety is that they are stated in terms of the marginal benefit with uncertainty, as defined in Definition Theorem 2.2 (Characterisation of Equilibrium) Symmetric equilibria under uncertainty of a concave game ; may be characterised as follows: a) MB ( ^m; ^º) > 0 > MB ( ^m +1; ^º) is a necessary condition for ^º to be an equilibrium, in which all play strategy ^m; moreover if supp ^º = f ^mg this condition is also sufficient; b) MB (1; ^º) 6 0 (resp. MB (m ; ^º) > 0) is necessary for ^º to be a symmetric equilibrium in which, all play strategy 0 (resp. m ); moreover if supp ^º = f0g (resp. supp ^º = fm g) this condition is also sufficient; c) MB ( ^m +1; ^º) =0isanecessary condition for ^º to be a symmetric equilibrium in which, f ^m; ^m +1g is the set of best responses, moreover if supp ^º = f ^m; ^m +1g this condition is also sufficient. Remark 2.3 The main assumption needed for this result is that each player s utility be concave in his/her own strategy. It holds regardless of the nature of the strategic interactions. Indeed it even holds in the absence of aggregate externalities. However these factors will affect the functional form of MB and hence the equilibrium strategies. 3. FREE RIDING UNDER UNCERTAINTY In this section we present our model of public good provision under uncertainty. We have chosen a relatively simple model. This enables us to focus on the effects of uncertainty. 3.1 The Model There are two goods, a public good Y and a private good X. There are n individuals. Each has utility function u i (y;x i )=w(y) dx i,wherey denotes the level of public good provision and x i denotes individual i s contribution to the public good (in terms of private good). 4 Contributions may only take integer values in the range 0 6 x i 6 m. Thus each player has a finite set of pure strategies. We make this assumption since, at present, equilibria under uncertainty have been more 4 There are a number of possible extensions to this basic model which one might wish to consider. In particular it has been suggested to us that the assumptions that utility is linear in the private good and separable between the two goods should be relaxed. In the more general model in section 6, neither of these assumptions is made. As can be seen, most of our conclusions remain valid in the more general model. 14

16 extensively studied in games with finite strategy sets. We believe that, in principle, it is possible to extend our results to the case where contributions are continuous. Individuals are assumed to have a sufficiently large endowment that they are able to contribute m. The maximum total amount that any one player believes his/her opponents can donate is (n 1)m. The level of public good provision is given by the production function, y = F ( P n i=1 x i). Notation 3.1 Define G : R! R by G(x) =w(f (x)) and r = G(r) G(r 1): By definition, r is the marginal benefit (in utility terms) of contributing the rth unit to the production of the public good. Throughout this section, we shall make the following assumption. Assumption 3.1 (Concavity) The function G is strictly concave and G(0) = 0: The function G; measures the benefit, in utility terms, of contributions to the public good. Provided w and F are concave, it will be strictly concave if either there is diminishing marginal utility of the public good (w is strictly concave) or there are decreasing returns to scale (F is strictly concave). It is possible to allow for increasing returns to scale in production, provided that these are offset by diminishing marginal utility. The public goods model is a concave game. It is a game of aggregate externalities. To see this define f : S i! R by f (x i )= P j6=i x j: There are positive spillovers, since any given player s utility is raised when his/her opponents donate more. Moreover it is a game with strategic substitutes, since higher contributions of others increase the supply of the public good, which reduces your marginal utility of the public good and hence the marginal benefit of your own contributions. Lemma 2.1 shows that, in equilibrium, the perceived marginal benefit is a weighted average of marginal benefits at a number of contribution levels for the opponents between 0 and the equilibrium level. When G is concave, this raises the perceived marginal benefit and hence the equilibrium contributions compared to the model without uncertainty. 15

17 3.2 Comparative Statics We shall now investigate comparative statics by changing uncertainty while keeping other factors constant. Assume that the following condition holds: n ¹m >d> n ¹m+1 : (1) By Theorem 2.2, this implies that in equilibrium without uncertainty, all would contribute ¹m. We shall investigate comparative statics by parametrising uncertainty. Consider a class of symmetric equilibria, which are continuously parametrised by a real number, such that =0corresponds to no uncertainty (additive beliefs) and =1corresponds to complete uncertainty (maximin preferences) and if ^ >¹ then º ^ is more uncertain than º ¹. (The term more uncertain is formally specified in Definition 3.1 below.) Let us consider how the symmetric equilibrium changes as increases from 0 to 1. For close to 0, the equilibrium is unchanged and all continue to contribute ¹m units. As increases this is replaced by a mixed equilibrium in which levels ¹m and ¹m +1are both best responses. Further increasing will result in an equilibrium in which all contribute ¹m+1 units. As continues to be increased we alternately get pure and mixed equilibria with successively higher levels of contributions. Eventually there will be complete uncertainty i.e. =1. In this case all will contribute n ¹m units and the total contribution to the public good will be n 2 ¹m: If n is large, this illustrates quite spectacularly how uncertainty can increase provision of a public good. From Theorem 2.2, we see that MB (m; º) > d > MB (m +1;º) is the condition for an equilibrium in which all contribute m units. By construction, in the limit as tends to one, º(A) = 0;A 6= S i. Intuitively, as uncertainty increases, confidence in the contributions of others falls. As a consequence, the weight assigned to such contributions in the Choquet integral also falls. In the limit, this weight tends to zero. Lemma 2.1 implies that the equilibrium condition becomes m0 > d > m0+1; comparing with equation 1 we see that m 0 = ¹mn, provided n ¹m 6 m. This may be explained as follows. Without uncertainty, equilibrium occurs where the marginal 16

18 x 2 0 x 1 Figure 1: =1 benefit (of the total contribution) first falls below d. This occurs where the total contribution is n ¹m. Under complete uncertainty the individual loses all confidence in the contributions of others and only assigns any weight to his/her own contribution. Again each individual contributes until the marginal benefit falls below d and hence (s)he chooses contribution n ¹m. Definition 3.1 We say that ^º is more uncertain than ¹º if for all non-empty A $ S i ; ^º(A) + ^º(:A) < ¹º(A)+¹º(:A). 5 The strict inequality is needed to provide unambiguous comparative statics. Note that if ^º is more uncertain than ¹º; then both the maximal and minimal degrees of ambiguity of ^º are greater than those of ¹º. The comparative static properties of the model follow from Propositions 6.4 and 6.2. If players beliefs can be represented as simple capacities, these results can be given a slightly different interpretation. Recall that a simple capacity has the form º = ¼, where¼ is interpreted as a probability estimate and is a degree of confidence in that estimate. Our existence proof implies that an equilibrium in simple capacities, will exist for any given degree of confidence. In this case, the comparative static result can be interpreted as saying that a reduction in the exogenous degree of confidence will increase the equilibrium contributions. In 5 A similar definition has been proposed by Marinacci [36], who built on earlier work by Dow and Werlang [17]. 17

19 x 2 0 x 1 Figure 2: = 1 2 other words, the more uncertainty there is, the less free riding we would expect. The comparative statics of changing the level of uncertainty are illustrated by figures 1 and 2, which assume that there are two players. Each diagram shows the indifference curves of player 1, in x 1 x 2 space. We illustrate the case where G is exponential. Figure 1 shows the case of no uncertainty. As usual, the indifference curves are U-shaped and the reaction function of player 1 is downward sloping with slope -1. In figure 2 there is uncertainty. As uncertainty about player 2 s contribution increases, it becomes less important to player 1. Thus a unit of player 1 s own contribution becomes worth more to him/her in terms of player 2 s contributions. Hence the indifference curves become steeper. For similar reasons, changes in player 2 s contribution have smaller effects on player 1 s optimal contribution, hence player 1 s reaction function is steeper when there is uncertainty. The intercepts of the reaction functions remain unchanged while their slopes increase, which implies that equilibrium provision of the public good increases with increases in uncertainty. 3.3 Pareto Optimality In this section we discuss conditions under which public good provision under uncertainty will be Pareto optimal (henceforth PO). In a conventional (Nash) equilibrium, provision is sub-optimal 18

20 since individuals do not take into account benefits to others when deciding how much to contribute. Uncertainty has the effect of changing the perceived marginal benefit of donations. However it is still the case that individuals ignore benefits going to others. Thus while uncertainty increases public good provision, there is no reason to believe that it will increase by the right amount to achieve Pareto optimality. Intuitively, since the determinants of contributions under uncertainty and the PO contribution level are different, one would expect that it is possible to have either too much or too little of the public good. One can show, by example, that this intuition is correct. As shown above, equilibrium contributions are increasing in the level of uncertainty. This implies that, if the contribution level under complete uncertainty is greater than the PO level, there is some level of uncertainty for which the equilibrium contribution level will be PO. Whether or not the contribution under complete uncertainty will be greater or less than the PO level, depends upon the functional form of G. For a power function it is less than the PO level. If G is exponential, the complete uncertainty contribution level may be greater or less than the PO level, depending on the parameter values. It is greater than the PO level for more concave functions of the exponential form. This is to be expected, since uncertainty raises the marginal benefit of contributions by a greater amount the more concave is G: If G is logarithmic, the PO contribution is exactly equal to that, which would be obtained under complete uncertainty. Note that by the Pareto optimal contribution level, we mean that level of contributions which is Pareto optimal in the absence of uncertainty. (In this model there is a unique PO level of public good.) This is an ex-post notion of Pareto optimality. 4. INCREASING RETURNS TO SCALE 4.1 Preliminaries In this section we investigate how our results are modified if G is convex rather than concave. We find that in this case, uncertainty can hinder the voluntary provision of public goods. With convex G, marginal benefit is greater, the higher the current level of contributions. When there 19

21 is uncertainty, any given individual is less able to rely on the contributions of others. Hence uncertainty reduces the anticipated marginal benefit of a contribution. These results clarify the circumstances in which uncertainty is likely to increase contributions. Assumption 4.1 (Convexity) G is strictly convex and G(0)=0. Throughout this section we shall require Assumption 4.1 to hold. As already noted we do not need to assume concavity for our existence result, Theorem 2.1. Hence it continues to hold if G is convex. The following lemma shows that, with convexity, each individual will either make no contribution or contribute the maximum possible amount. Lemma 4.1 If Assumption 4.1 (Convexity) is satisfied, then the set of best responses for individual i is a subset of f0;m g. Note that the proof of Lemma 4.1 applies whether or not º is additive, hence it holds with or without uncertainty. 4.2 Without Uncertainty To set a bench mark, we shall characterise the Nash equilibria (without uncertainty) when there are increasing returns to scale. Notation 4.1 Define ± r = G(r + m ) G(r). Hence ± r is the extra benefit of contributing m units rather than 0, given that the total contribution by other individuals is r. Note in the case of convex G, the optimal contribution is determined by the difference in total benefit between contributing m ratherthan0andnotintermsofmarginal benefits. Assumption 4.1 implies that ± r is strictly increasing in r. The following result characterises equilibrium without uncertainty. There are three possible structures for the set of equilibria, depending on the marginal cost of contributions, d. In the first, it is a dominant strategy to make the maximum contribution, m. In the second there are multiple equilibria. Roughly an individual wishes to contribute m if others do so, but would contribute 0 if (s)he expects others to contribute 0. In this range, the public goods model is effectively a coordination game. There are two pure strategy equilibria, one of which Pareto dominates the 20

22 other. Finally, when the marginal cost of contributing is high, it is a dominant strategy for all individuals to contribute 0. Recall (n 1)m, is the largest amount which all individuals other than j could contribute. Theorem 4.1 Characterisation of equilibrium. If there are increasing returns to scale (i.e. Assumption 4.1 is satisfied), symmetric equilibrium without uncertainty may be characterised as follows: a) if ± (n 1)m >± 0 >m d; the only equilibrium contribution level is m ; b) if ± (n 1)m > m d > ± 0, there exist two pure strategy equilibria, one in which all contribute 0 and one in which all contribute m ; in addition there is a mixed strategy equilibrium in which all individuals randomise between contributing m and 0; c) if m d>± (n 1)m >± 0, the only equilibrium contribution level is 0. Remark 4.1 Since the individual is deciding whether to contribute m or 0 the marginal cost d of contributing is not relevant. What is relevant is the total cost of contributing which is equal to m d. 4.3 Comparative Statics In contrast to the case of decreasing returns to scale, if G is convex, uncertainty is a barrier to voluntary provision of public goods. In the equilibrium with uncertainty, contributions are less than or equal to those without uncertainty. This is demonstrated by the results in this subsection. The first of these, shows that there is a unique equilibrium without uncertainty, this is also the symmetric equilibrium with uncertainty. Proposition 4.1 Assume ± M >± 0 >m d (hence without uncertainty only the high contribution equilibrium exists). Then in any symmetric equilibrium with uncertainty the only equilibrium contribution level is m. Remark 4.2 For these parameters, without uncertainty, any given individual would wish to contribute m, whatever others do. Hence it is not too surprising that the optimal contribution with uncertainty is m. More formally this result holds because CEU preferences respect dominance. The following proposition shows that, in almost all cases, where both equilibria exist without uncertainty, only the Pareto inferior equilibrium (without uncertainty) will arise if there is enough ambiguity. Proposition 4.2 Assume ± M > m d>± 0 then there exists ¹ such that if the minimal degree of ambiguity is > ¹, the equilibrium strategies are unique and involve all individuals making zero contribution to the public good. 21

23 The next result shows that, if only the low contribution equilibrium exists without uncertainty, the same is true with uncertainty. The proof is omitted since it is similar to that of Proposition 4.1. Proposition 4.3 Assume m d > ± M >± 0, then in any symmetric equilibrium with uncertainty, the only equilibrium contribution level is 0. The case where there are multiple equilibria without uncertainty, is the most interesting. In the other two cases, each individual has a dominant strategy. Thus it is not very surprising that the equilibria are unaffected by uncertainty. If there are sufficiently strong increasing returns to scale, uncertainty will reduce voluntary donations. Note that this result was obtained under the assumption that G is convex. It is not sufficient that there be increasing returns to scale in production, it is required that increasing returns to scale be sufficiently strong to offset diminishing marginal utility of the public good. When G is concave, increases in uncertainty caused a gradual increase in contribution levels. In contrast with convex G, increases in uncertainty either have no effect or cause a catastrophic collapse in contributions. One piece of evidence in favour of these results, is that there are instances where volunteer armies have been formed (see, for instance, Orwell [43] ). However, we are not aware of any examples of volunteer navies or airforces. Armies are subject to approximately constant returns to scale (most volunteer armies would mainly consist of infantry with rifles), which with diminishing marginal utility, would cause G to be concave. In contrast there are sharply increasing returns to scale in air forces and navies, hence G cannot be assumed to be concave in these cases. 5. EXPERIMENTAL EVIDENCE There is a large experimental literature on public goods, for surveys see Davis and Holt ([15], Ch. 6) and Ledyard [34]. We shall only comment on those aspects of the literature which appear to be relevant to Knightian uncertainty. The following stylised facts emerge from experimental 22

24 research. 1. There is no significant evidence of free-riding in single shot games. 2. When subjects play a repeated game, provision of the public good decays toward the freeriding level with each repetition. 3. Experienced subjects free-ride more than inexperienced subjects (Isaac, Walker and Thomas [30] ). 4. Face to face communication reduces free-riding (Hackett, Schlager and Walker [28] ). It is tempting to interpret these results in terms of uncertainty. We have already shown uncertainty can significantly reduce free-riding. It does not seem unreasonable to assume that uncertainty is highest in a one-shot game and that it is reduced by repetition. (Camerer and Karjalainen [10] interpret the effect of uncertainty in repeated games similarly, see especially p. 348.) Hence we can also explain the increase in free-riding over time. Likewise one would expect experienced subjects to perceive less uncertainty, hence fact 3 is also compatible with our model. Of the experimental evidence only the result that face to face communication reduces free-riding fails to support our theory. While communication is likely to reduce uncertainty, we believe it may have a number of other effects such as establishing a focal point, creating feelings of loyalty etc. Hence we do not believe this evidence is very damaging, since the reduction in uncertainty is confounded with other factors, which have opposing effects. The main previous explanation of the observed low level of free riding is altruism. (See, for instance, Andreoni [2].) However altruism does not explain why repetition decreases freeriding. This has been explained by postulating that subjects learn about the structure of the game with repetition. (Note that this is not incompatible with our explanation in terms of uncertainty.) In our theory, the uncertainty concerns the opponents play. If subjects play the same game with new opponents there is not likely to be a great decrease in uncertainty, however they would learn about the structure of the game. A relevant experiment is reported by Andreoni [1], who divides subjects into two groups called partners and strangers. The partners group plays a public goods game ten times with the same group of five players, i.e. partners play a conventional repeated game. The strangers 23

25 group plays ten rounds of the same game, except any given subject is playing the game with different opponents on each round. In other words, strangers play a one-shot game ten times. The hypothesis being tested is that subjects build a reputation for cooperation in the repeated game along the lines described by Kreps, Milgrom, Roberts and Wilson [33]. If this hypothesis were correct it would imply that partners are more likely to cooperate since they have more scope for reputation building. In fact it was found that strangers were more likely to cooperate. This is compatible with our model. It seems reasonable to assume there is more uncertainty in the strangers group, since players were not able to get used to the behaviour of their opponents. In contrast, it does not seem likely that altruistic feelings should be stronger towards strangers than partners. 6 In a second experiment Andreoni [1], investigates the effect of unexpectedly restarting an experiment. Subjects were told that they would play ten rounds of a public goods game. After the tenth round the experiment was restarted. The subjects were told that they would play an additional ten rounds of the same game. The results were that during the first ten rounds, free riding slowly increased as in other experiments. The first round after restarting had a low level of free-riding similar to first round of the initial experiment. These results appear to be difficult to explain by conventional models of learning, since such models would predict that the first round after the restart would be like the 11th round of a longer experiment. It is possible that the unexpected restart created uncertainty. In which case, the experiment would be compatible with our results. 7 Andreoni argues that his results can only be explained by theories of non-standard behaviour. Although he does not explicitly consider Knightian uncertainty, we believe the present paper is in the spirit of this suggestion. Van Huyck, Battalio and Beil [31] test a special case of our model, where the production function is min fx 1 ; :::; x n g. This can be viewed as a limiting case of the model with increasing 6 Although Croson [14] failed to replicate Andreoni s result, we believe thatthe subsequent experimental literature is broadly supportive of our conclusion that this result is due to uncertainty. A survey of this literature can be found in Andreoni and Croson [3]. 7 In a subsequent experiment Croson [14] has confirmed the restart effect. 24