Collective funding for the protection of the global commons underasymmetric benefits
Hiroaki Chiba
Nishimura & Asahi

Abstract
The challenges associated with managing shared resources have frequently been highlighted ineconomic scholarship. This paper examines a particular form of cooperative mechanism aimed atencouraging the investment of greater effort in the preservation of shared resources in the context ofspace debris mitigation, namely, a collective fund dedicated to the promotion of mitigation efforts.
Adopting a simple two-player game theoretic model that considers the existence of asymmetry inbenefits accruing to the potential benefactors, this paper demonstrates not only that such a fund would
not attract optimal funding levels but also that the players’ only sensible response would be that,irrespective of the degree of asymmetry in benefits, all contributions should come from the player who
receives the relatively larger benefit from the shared resources while the other player should contributeno funding at all. This implies that a collective fund dedicated to the promotion of mitigation efforts
would struggle to attract contributions from actors who derive relatively fewer benefits from theutilization of outer space.

1.

Introduction

Owing to the pursuit of self-interest by individual members of society, shared resources tend to beoverexploited, leading to suboptimal consequences for society as a whole, a situation often referred toas the “tragedy of the commons” [4]. The management of the global commons, which areinternationally shared resources, face particular challenges due to the absence of a supranational
authority that can compel the relevant actors to behave in socially optimal ways.
Earlier studies have applied game theoretic models to analyze the possibility of cooperation inpreserving such global commons as clean air and outer space. Some of these studies have considered
the presence of asymmetry in benefits and costs among different actors [1,8,10]. This paper likewise
employs a game theoretic approach to examine a particular form of cooperation that has not been
analyzed in earlier literature—a collective fund dedicated to protecting the global commons.
The idea of establishing regional or global funds has recently been proposed in the context ofspace debris mitigation [9]. Space debris comprises disused human-made objects in space, including
abandoned satellites and launch vehicle parts. Pieces of space debris may collide with and disable
functioning spacecraft, thereby posing a significant threat to space activities. Countless pieces ofdebris are already orbiting Earth (as of February 2020, ESA estimates 34,000 objects larger than 10

cm, 900,000 objects ranging between 1 cm and 10 cm, and 128 million objects ranging between 1 mmand 1 cm [2]), and the number is increasing. Donald J. Kessler and Burton G. Cour‐Palais notably
predicted that the space debris population will begin to grow exponentially at some point in aphenomenon termed “Kessler syndrome” [6,7]. To mitigate the threat posed by space debris and avoid
potentially catastrophic consequences, the proposed funds are intended to subsidize clean launches
and debris removal activities [9].

2.

Methods

This paper employs a game theoretic model in which the potential benefactors are players who try tomaximize their utilities. The players’ utilities are represented by a set of utility functions, and theplayers’ behavior is analyzed by finding the Nash equilibrium, a standard solution concept in gametheory whereby all actors employ their mutual responses and no actor can gain from individualdeviation [3].
The game has two players, both of whom are potential benefactors of a collective fund dedicatedto the protection of the global commons, specifically, outer space in this context. Although there arelikely to be more than two potential benefactors for an international or regional fund for space debris
mitigation, restricting the number of players will help to simplify the analysis without fundamentallyaffecting the results and implications of the analysis. The two players may either be interpreted as twoindividual states or two separate groups of states. Each player chooses a non-negative real value (𝑐1 ∈ℝ+, 𝑐2 ∈ ℝ+), which is the amount of respective contribution to the fund. Reflecting the propertiesof collective funds, contributions of the same amount from different countries are assumed to carry
the same weight. Consistent with this assumption, the fund’s total size would simply be the sum of c1and c2. The benefit produced by the global commons is represented by a positive real value (𝑝 ∈ ℝ++)
which is common to both players, but the benefit gained by one of the players (player 2) is multiplied
by a discount factor (𝛿 ∈ (0, 1]). This factor is introduced to capture the real-world situation whereby
the exploitation of outer space is subject to significant technological constraints and differences intechnological capabilities result in differences in the benefits derived from outer space utilization.
Although any state or corporation can potentially take advantage of advanced technologies byacquiring them from states and corporations that have such technologies, in reality, trade in products
and technologies relating to outer space activities is rigorously controlled by national governments,
primarily owing to national security and diplomatic concerns [5]. Even when they are to be traded, thegovernments or relevant industries of technologically advanced states can derive profit throughexercising their market power. It is assumed that a complete absence of contribution would lead to acatastrophic event whereby the global commons would be entirely depleted. In this case, the benefit
produced by the global commons (p) would be zero. This corresponds to a real-world situation where,
for example, a specific range of orbit might become practically unusable due to the presence ofoverwhelming amounts of space debris, perhaps due to Kessler syndrome having been triggered. By
increasing the contribution to the collective fund (c1, c2), the players can reduce the probability of suchoutcomes to zero at maximum, but with diminishing marginal returns.
The utility functions of player 1 (u1) and player 2 (u2) capture the above settings.
𝑢1 (𝑐1 , 𝑐2 ) = −𝑐1 + (1 − 𝑒 −𝜆(𝑐1 +𝑐2 ) )𝑝𝑢2 (𝑐1 , 𝑐2 ) = −𝑐2 + (1 − 𝑒 −𝜆(𝑐1 +𝑐2 ) )𝛿𝑝
The common coefficient of p and δp that appears in both u1 and u2 takes the form of the
cumulative distribution function of the exponential distribution, which captures a situation in which
uncertainty exists regarding the amount of collective investment that would be sufficient to avoid
catastrophic consequences. The uncertainty may arise from the difficulty in accurately estimating andpredicting the status of space debris as well as the difficulty in eliminating the possibility of making
accidental addition to space debris even when an appropriate mitigation measure is in place. While theexponent in the cumulative distribution function is often interpreted as the waiting time until the event
occurs, with a longer waiting time leading to a higher probability that the event will occur, it isinterpreted in this model as the level of total contribution to the fund in this model, with a greater
contribution leading to a higher probability that the catastrophic consequence will be successfullyprevented. The higher probability that the catastrophic consequence may be avoided can alternatively
be interpreted as the higher level of utility gained from space utilization, associated with the lower
probability that functioning spacecraft will collide with space debris and the lower cost of making
spacecraft robust against space debris. λ is an exogenous positive real parameter that allows foradjustment with respect to the fund’s effectiveness in preventing catastrophic consequences. Althoughe, the base of natural logarithm, is selected as the base here, any real base larger than 1 other than e
can be represented by adjusting λ by multiplying the exponent by the natural logarithm of the real basewithout affecting the analysis.

3.

Results
The partial derivatives of u1 and u2 with respect c1 and c2, respectively, are𝜕𝑢1= −1 + 𝜆𝑒 −𝜆(𝑐1 +𝑐2) 𝑝𝜕𝑐1
𝜕𝑢2= −1 + 𝜆𝑒 −𝜆(𝑐1 +𝑐2 ) 𝛿𝑝𝜕𝑐2
Accordingly, the optimal contribution levels for player 1 and player 2 are as follows:𝑐1∗ =log 𝜆𝑝
log 𝜆𝑝− 𝑐2 𝑖𝑓 𝑐2 ≤𝜆𝜆𝑐1∗ = 0 𝑖𝑓 𝑐2 >𝑐2∗ =log 𝜆𝑝𝜆log 𝜆𝛿𝑝log 𝜆𝛿𝑝− 𝑐1 𝑖𝑓 𝑐1 ≤𝜆𝜆𝑐2∗ = 0 𝑖𝑓 𝑐1 >log 𝜆𝛿𝑝
𝜆When 𝛿 = 1, both player 1 and player 2 optimally should contribute to the fund until the total
contribution amount reaches log 𝜆𝑝⁄𝜆. The Nash equilibrium solution does not reveal how muchplayers 1 and 2 would each contribute. However, when 0 < 𝛿 < 1, irrespective of the magnitude ofδ, the only Nash equilibrium is that player 1 contributes log 𝜆𝑝⁄𝜆 and player 2 contributes zerobecause log 𝜆𝛿𝑝⁄𝜆 < log 𝜆𝑝⁄𝜆.
Taking the partial derivative of the total utility of player 1 and player 2 with respect to 𝑐1 + 𝑐2,
it can be shown that the aggregate contribution to the fund would always be lower than the socially
optimal level, because log 𝜆𝛿𝑝⁄𝜆 ≤ log 𝜆𝑝⁄𝜆 < log 𝜆(1 + 𝛿)𝑝⁄𝜆 always hold.

(𝑐1 + 𝑐2 )∗ =4.

log 𝜆(1 + 𝛿)𝑝
𝜆

Discussion

The model analyzed in this paper suggests that a collective fund for the protection of the globalcommons, including outer space, is unlikely to attract the optimal level of total funding, which isconsistent with earlier studies that predicted that collective action would yield suboptimal results.
Furthermore, this papers’ findings reveal a unique Nash equilibrium under such a mechanism whenbenefits are asymmetric, whereby the actor receiving the relatively higher benefit makes the entire
contribution to the fund while the other actor contributes nothing. The intuitive explanation for thisresult is that, as the actor receiving the higher benefit would contribute a greater amount than is optimalfor the actor receiving the lower benefit, regardless of the amount that the latter contributes, they willalways be better off contributing less.
This result implies that a collective fund for the protection of the global commons would facesignificant difficulty in enticing contributions from actors who derive relatively fewer benefits fromthe commons. In practice, such asymmetry may arise when access to the commons is subject totechnological constraints, as is the case for outer space. It may be predicted from the model that theinitiative to create a collective fund for space debris mitigation would inevitably have to be led by agroup of technologically advanced countries that benefit most from the utilization of outer space andthat the funding would predominantly, if not entirely, come from these countries.
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