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Morally monotonic choice in public good games

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Abstract

Rational choice theory, including models of social preferences, is challenged by decades of robust data from public good games. Provision of public goods, funded by lump-sum taxation, does not crowd out private provision on a one-for-one basis. Provision games elicit more of a public good than payoff-equivalent appropriation games. This paper offers a morally monotonic choice theory that incorporates observable moral reference points and is consistent with the two empirical findings. The model has idiosyncratic features that motivate a new experimental design. Data from our new experiment and three previous experiments favor moral monotonicity over alternative models including rational choice theory, prominent belief-based models of kindness, and popular reference-dependent models with loss aversion.

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Notes

  1. Andreoni (1995) refers to these two game forms as “positively-framed” and “negatively-framed” public good games.

  2. For literature reviews, see the early work by Ledyard (1995) and the selective survey by Chaudhuri (2011). Our Online Appendix I.1 provides another selective survey.

  3. See, for examples: Andreoni (1995), Sonnemans et al. (1998), Park (2000), Messer et al. (2007); Fujimoto and Park (2010); Bougherara et al. (2011), Cubitt et al. (2011), Dufwenberg et al. (2011); Cox et al. (2013); Fosgaard et al. (2014), Cox (2015), Cox and Stoddard (2015), Khadjavi and Lange (2015), and van Soest et al. (2016).

  4. See, for examples: Abrams and Schmitz (1978, 1984), Clotfelter (1985), Kingma (1989), Andreoni (1993), Bolton and Katok (1998), Khanna and Sandler (2000), and Ribar and Wilhelm (2002). For incomplete crowding out with distortionary taxes see Bernheim (1986).

  5. By “conventional rational choice theory” we mean consequentialist rationality (e.g., Samuelson, 1938; Chernoff, 1954; Arrow, 1959; Sen, 1971, 1986, 1993) and its prominent special cases including conventional preference theory (e.g., Debreu, 1959; Hicks, 1946; Samuelson, 1947; textbooks), revealed preference theory (e.g., Afriat, 1967; Varian, 1982; textbooks) and (unconditional) social preferences models (e.g., Andreoni & Miller, 2002; Bolton & Ockenfels, 2000; Cox & Sadiraj, 2007; Fehr & Schmidt, 1999).

  6. These and other implications of these models are explained in Sects. 2 and 3.

  7. See Cox, et al. (2019) for applications to dictator games (non-strategic environments).

  8. We define “non-binding” lower bound as one that is smaller than previously-observed allocations of all group members in the full game as well as their reported beliefs about the others’ choices.

  9. Popular models of (consequentialist) social preferences belong to conventional rational choice theory because they are characterized by utility functions defined over final (monetary) payoffs that provide complete and transitive orderings in the payoff space.

  10. The scarce resource in our experiment consists of 10 “tokens” that can be allocated between private and public accounts, with each token worth $1 in the private account and $1.5 in the public account that is equally shared by two players (i.e., the marginal per capita return is 0.75). In both appropriation and provision games, each player’s final payoff is the sum of tokens in the private account and 0.75 times the total number of tokens in the public account.

  11. In the provision game, with endowments of 10 tokens in each private account, my feasible set, in (own payoff, other’s payoff)-space, is \(X = \{ (10 - x + .75(5 + x),\;10 - 5 + .75(5 + x))|x \in \{ 0, \cdots ,10\} \}\) when the other player contributes (or so I believe) 5. If I contribute 7 to the public account then my choice set is the singleton \(X^{*} = \{ (12,14)\} .\) Generally, an agent’s choice set may not be a singleton, but we work with singleton choice sets in examples so the illustration of ideas is simple.

  12. Even with a stronger conventional assumption, such as GARP, choice sets are not generally singletons.

  13. See also Samuelson (1938), Chernoff (1954), Sen (1993).

  14. For an example, Sect. 3.3.1, which reports (observable) four-dimensional reference points (two dimensions for each player).

  15. More generally, if the moral reference point has four dimensions (i.e., two dimensions for each player), then \(t^{1} ,s^{1} \in {\mathcal{R}}^{4}\) and the expression is \((t_{11}^{1} - s_{11}^{1} ) + (t_{12}^{1} - s_{12}^{1} ) = 0 = (t_{21}^{1} - s_{21}^{1} ) + (t_{22}^{1} - s_{22}^{1} )\).

  16. Here, the moral reference point has m dimensions per player. If it has only one dimension per player, as in the special case in the paragraph above, then \(m = 1\).

  17. As an illustration for applied research, in Online Appendix I.4, we apply a parametric special case of \(U( \cdot )\) − where the weights \(w_{k} ( \cdot )\) are normalized natural exponential functions of the moral reference point − to data from two previous experiments (Andreoni, 1995; Khadjavi & Lange, 2015) and data from a new experiment reported herein.

  18. We will explain our specification of moral reference point in Sect. 3.

  19. Example 2.a is very similar to one suggested by an anonymous reviewer. Let \(b_{2}\) and \(c_{1}\) denote player 1’s first- and second-order beliefs. For the Dufwenberg and Kirschteiger (2004) model, player 1’s utility in the full game is \(\pi_{1} (g_{1} ,b_{2} ) + Y\gamma^{2} (g_{1} - 5)(b_{2} - 5)\), and \(\pi_{1} (g_{1} ,b_{2} ) + Y\gamma^{2} (g_{1} - 9)(b_{2} - 9)\) in the contraction game with low bound of 8, where \(\gamma = .75\) and \(g_{1}\) is player 1’s contribution. A player 1 with sensitivity parameter, \(Y > 0.15\), first-order belief, \(b_{2}\) = 8 (and any second order belief) is a full contributor (\(g_{1}\) = 10) in the full game but a “free rider” (\(g_{1}\) = 8) in the contraction game.

  20. At optimal interior choice, \(\pi^{s}\) in problem \((S,s)\), \(u^{\prime}(\pi_{1}^{s} )/u^{\prime}(\pi_{2}^{s} ) = (\theta_{2}^{s} /\theta_{1}^{s} )/(1/\gamma - 1) < (\theta_{2}^{t} /\theta_{1}^{t} )/(1/\gamma - 1)\) where the equality follows from optimality of \(\pi^{s}\) and the inequality follows from t favoring 2 and disfavoring 1 compared to s. Hence, \(\pi_{2}^{s}\) is too small to be optimal in problem,\((T,t)\), which implies that player 1’s allocation must be larger,\(g_{1}^{t} (g_{2} ) > g_{1}^{s} (g_{2} ).\)

  21. Assume additively separable preferences, with utility of consumption \(v(\pi ) = \sum {u_{i} (\pi_{i} )}\), and for each dimension i, gain/loss utility \((\pi_{i} - r_{i} )\) for gains, and \(\lambda (\pi_{i} - r_{i} )\) for losses (with \(\lambda > 1\)). A player 1 that allocates 3 in the provision game, reveals (*) \((v_{1} + 1)(\gamma - 1) + (v_{2} + 1)\gamma = 0.25( - v_{1} + 3v_{2} + 2) = 0.\) In the appropriation game, her marginal utility at allocation 3, is \((v_{1} + 1)(\gamma - 1) + (v_{2} + \lambda )\gamma = 0.25( - v_{1} + 3v_{2} + 3\lambda - 1) = 0.25(3\lambda - 3) > 0\) where the second equality follows from (*) and the inequality follows from loss aversion, \(\lambda > 1.\) Hence, player 1’s allocation in the appropriation game must be larger than in the provision game.

  22. Note that (*) \(\theta_{2}^{a} /\theta_{1}^{a} = \theta (15)/\theta (\sigma 15) = \left( {\theta_{2}^{p} /\theta_{1}^{p} } \right)\left( {\theta (5)/\theta (\sigma 5)} \right) < \theta_{2}^{p} /\theta_{1}^{p}\) for the choice function below Proposition 2. At optimal, \(\pi^{p}\) in the provision game, \(u^{\prime}(\pi_{1}^{p} )/u^{\prime}(\pi_{2}^{p} ) = (\theta_{2}^{p} /\theta_{1}^{p} )/(1/\gamma - 1) > (\theta_{2}^{a} /\theta_{1}^{a} )/(1/\gamma - 1),\) where the equality follows from optimality of \(\pi^{p}\) and the inequality follows from (*). So, the appropriation game requires a smaller left hand side ratio, hence, player 1’s allocation must be smaller,\(g_{1}^{a} (g_{2} ) < g_{1}^{p} (g_{2} ).\)

  23. Formal arguments for n-player games for conventional rational choice and morally monotonic choice are reported in Online Appendix I.2.

  24. In a provision game, a required minimum contribution,\(c > 0,\) produces a contraction. Government contribution to a public good financed by lump sum taxation is one way of implementing such a contraction. In an appropriation game, a contraction corresponds to a quota on maximum extraction, \(t > 0.\) The two types of contractions are payoff equivalent when \(c = W - t\).

  25. Note that if the lower bound, c is binding then by construction individual allocations are weakly increasing in c. For example, c = 2 (as in our illustration) is binding for player 1 if she chooses to allocate 1 in the public account in the full game but cannot do so in the contraction game.

  26. Here \(C = \{ c, \cdots ,W\} ,c > 0\) is non-binding if \(0 < c < \min (g_{i}^{b} ,\min (g_{ - i} ))\) where \(g_{i}^{b}\) is the smallest best response allocation of player i in the full game (i.e., c = 0).

  27. We have dropped subscripts on \(Y,\) \(\kappa\) and \(\lambda\) to simplify notation.

  28. In Fig. 1, \(r^{p} = (10,10)\) and \(r^{a} = (15,15)\).

  29. Separate detailed explanations of moral reference points in provision, appropriation, and mixed games with contraction (c > 0) or without contraction (c = 0) can be found in Online Appendices I.2 and II.5.A.

  30. 11.25 = (10 − 10) + 0.75(5 + 10) and 8.75 = (10 – 5) + 0.75(5 + 0).

  31. 15 = 0 + 0.75(10 + 10).

  32. The definition of total change, \(\delta\) appears in the notation paragraph just above the definition of M-Consistency.

  33. Recall that \(C = \{ c, \cdots ,W\} ,c > 0\) is non-binding if \(0 < c < \min (g_{i}^{b} ,\min (g_{ - i} ))\) where \(g_{i}^{b}\) is the smallest best response allocation of player i in the full game (i.e., c = 0).

  34. We thank the editor for suggesting the Reuben and Riedl (2013) paper.

  35. In both URE and UUE treatments, the game is a provision game with mpcr of 0.5, and in each group of three players, one player is endowed with 40 tokens whereas the other two are endowed with 20 tokens. The high-endowment player can contribute all 40 tokens in UUE but only up to 20 in URE. The low-endowment players can contribute all 20 tokens in both UUE and URE treatments.

  36. These estimated effects are for “token” endowments. The implied dollar amounts are economically significant. For example, the -0.11 token coefficient for Andreoni’s data corresponds to -6.6 (= 0.11*60) tokens when endowment of the public account is changed from 0 to 60 tokens. So, with n = 5 and mpcr = 0.5, the payoff from the public account decreases by $16.50.

  37. For the high-endowment player, the effect of a non-binding upper bound on own contribution in URE is a larger own minimal expected payoff than in UUE. So, for any given contribution of the low-endowment player, moral monotonicity predicts the high type’s best response is lower in URE than in UUE. For the low-endowment player, the non-binding upper bound on high’s contribution has no effect on low’s moral reference point, so moral monotonicity predicts that for any given contribution of the high-endowment player, low’s best response is the same in the two treatments.

  38. The experiment was approved by the Institutional Review Board at Georgia State University.

  39. In a CBC session, the contraction sets used in the first C task are the same as in a preceding BCB session.

  40. Exceptions to the “$1 less” criterion are when observed allocations in the preceding task are at a corner amount of 0 or close to 10. In a BCB session, if either subject guessed 0 or allocated 0 to the public account in the first B task then the set in treatment C would be integers from [0,10]. If application of the “$1 less” criterion would have resulted in a set with fewer than three options (i.e., lower bound 8 or 9) the set of allocations for task C was {5, 6,…,10}.

  41. In terms of the number of tokens allowed to be allocated to the public account this set is \(\{ 3,4, \cdots ,10\}\).

  42. One decision in each of the 2-game, 5-game, and 8-game; the order of the tasks was randomized across subjects.

  43. The experiment was not pre-registered. All of the data from the experiment we conducted are used in the regression reported in Table 5; we collected no other unreported data.

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Funding

We are grateful to the editor and reviewers for comments and suggestions that motivated significant improvements in the paper. The National Science Foundation (grant number SES-1658743) and the Experimental Economics Center at Georgia State University provided research funding. The replication material for the study is archived at https://doi.org/10.5281/zenodo.7425943.

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  1. Experimental Economics Center and Department of Economics, Georgia State University, Atlanta, GA, USA

    James C. Cox & Vjollca Sadiraj

  2. Department of Economics, Arizona State University, Tempe, AZ, USA

    Susan Xu Tang

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Cox, J.C., Sadiraj, V. & Tang, S.X. Morally monotonic choice in public good games. Exp Econ 26, 697–725 (2023). https://doi.org/10.1007/s10683-022-09787-2

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