Master’s Thesis
The redistribution model of research funding
A game theoretical approach
Arthur Molenaar
Student number: 11923709 Date of final version: 15 August 2018 Master’s programme: Econometrics
Specialisation: Mathematical Economics Supervisor: dr. M. J. van der Leij Second reader: dr. M. A. L. Koster
Statement of Originality
This document is written by Student Arthur Molenaar who declares to take full responsi-bility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of comple-tion of the work, not for the contents.
Contents
1 Introduction 1
2 Research background 3
2.1 Current funding process . . . 3
2.2 Issues with the current system . . . 4
2.3 Proposed solutions . . . 4 3 Mathematical model 6 3.1 The game . . . 6 3.1.1 Nash equilibria . . . 7 3.1.2 Coalition-proof equilibria . . . 8 3.2 Personal preference . . . 11 3.2.1 Fairness . . . 12 3.2.2 Equilibria . . . 15 4 Discussion 19 5 Conclusion 21 Abstract
In this thesis, we analyze a proposed new model of research funding distribution from a game theoretical point of view. In this model, rather than applying for grants through writing research proposals, peers can distribute the funding freely among themselves. The main point of concern is that researchers will conspire and form a coalition to improve their own funding. We show that the game is indeed vulnerable to coalitions in general. However, as long as we assume that players consider fairness in the distribution to some degree, there is no possibility to form enforceable coalitions and the Nash equilibrium in the game is unique.
1 Introduction
The current scientific research cycle revolves around the process of writing research proposals, obtaining a grant from the government or other organizations, doing research, writing an article to be published in an academic journal, and starting over. From a societal and scientific point of view, the aim of researchers should be to advance scientific knowledge as efficiently as possible. Part of this goal is that scientific grants should be awarded to the most ‘valuable’ research (regardless of the metric used to assess value). In many cases, novel, counter-intuitive or otherwise notable hypotheses attract more attention than hypotheses that confirm current understanding. This introduces an incentive for researchers to investigate these kinds of hypotheses, in order for them to maximize the probability of obtaining a grant. (Bergstrom, Foster, & Song, 2016) These exotic hypotheses are more likely to be false than more conservative research topics, which increases the likelihood of a false positive result down the road (Smaldino & McElreath, 2016).
Another negative side effect of this process is that a lot of value for the researcher is created in obtaining the grant, compared to the actual research. This leads researchers to spend on average 19 % of their time allocated to research preparing research proposals, according to Link, Swann, and Bozeman (2008). This is of course not a desirable outcome, but is a necessary feature of the current system. The scientific funding process is essentially a screening and signalling game, where the funding party is trying to assess how the available budget is optimally distributed given limited and asymmetric information. Part of this game is that researchers spend significant amounts of time signalling (i.e. writing proposals), since otherwise they are unlikely to receive the desired grants (Gross & Bergstrom, 2018).
Earlier research has focused on possible improvements of the current model. Bollen, Crandall, Junk, Ding, and Börner (2017), for instance, have developed a system where peers distribute the available budget among themselves, where a fixed base amount is
research useful. Initial analyses using citations as a proxy for funding distribution found that a similar funding distribution can be obtained with a lot less effort—primarily, there is no need to write elaborate research proposals for every study. A disadvantage of this model is the possibility of ‘groupthink’ where researchers who represent the consensus get the majority of extra funding. On the other hand, since every researcher gets the base funding, there is an opportunity for junior researchers to develop their own fields of interest, relatively independent of the current popular fields of interest.
On the other hand, there is likely a collusion opportunity if the system is completely unrestricted, where a subset of players distribute their available funding among themselves to become relatively independent of the other players while assuring sufficient funding. The strategy players must develop to decide how to distribute funding among their peers is inherently a game theoretic question. In this thesis, we will develop an appropriate model to investigate the possible strategies and make recommendations regarding the system that should be used in order to minimize the possibility of undesired outcomes.
2 Research background
2.1 Current funding process
Research funding in the Netherlands is mainly obtained through The Netherlands
Organi-sation for Scientific Research (nwo), which has various funding instruments available for
different kinds of researchers. The funding process as outlined on its web site is standard in the international academic community, and fairly elaborate. It consists of multiple stages. (The Netherlands Organisation for Scientific Research [nwo], 2018)
1. Pre-proposal. A pre-proposal is necessary if funding demand far exceeds supply. In this case, applicants need to send a brief pre-proposal so that nwo can indicate whether funding is realistic before submitting a full proposal.
2. Application. Those interested in funding need to submit their application with a research proposal to the nwo for consideration. The nwo “first determines whether the research proposals satisfy the admissibility criteria for the funding instrument concerned.” (nwo, 2018)
3. Assessment. The research proposal will be sent out for peer review (both domestic and internationally). These external experts are able to express their opinion on the proposal. The nwo then appoints a committee that issues an advice as to which proposals should receive funding.
4. Decision. The board of nwo finally decides which projects receive funding, based on all available information. Under normal circumstances, the board will adopt the advice of the selection committee.
2.2 Issues with the current system
The system of writing research proposals, which are subsequently peer-reviewed, is very costly and time-consuming. As mentioned in Chapter 1, approximately 20 % of research time is spent drafting proposals (Link et al., 2008). This obviously does not include the peer-review work that has to be performed in the assessment phase of the process, in order to evaluate other proposals.
Moreover, according to Cole, Cole, and Simon (1981), there is a large component of chance whether or not an individual researcher receives funding, since there is a high variance in how different reviewers assess the same research proposal. In particular, they find that the costly and time-consuming peer review process only takes away about half of the randomness that would have been achieved if grants were assigned using coin tosses. The fact that research grants are assigned 50 % randomly is not necessarily bad. After all, it is impossible to eliminate all randomness, simply because different reviewers probably have different opinions on the direction the field of research should take and what constitutes good research. However, Bol, De Vaan, and Van de Rijt (2018) have found that there is a large difference in the career paths between researchers who received a grant early in their careers and those who did not. This is known as the Matthew effect, a term first coined by Merton (1973). According to Bol et al., “grant winners just above the threshold accumulate more than twice as much funding during the subsequent eight years as nonwinners with near-identical review scores that fall just below the threshold” (Bol et al., 2018, p. 4887). They suggest that early funding itself is an asset for acquiring later funding.
2.3 Proposed solutions
Different alternatives to the current funding process have been proposed. A benchmark approach would be to simply divide grants randomly, that is, a lottery. That model would obviously eliminate any bias that reviewers might have completely. On the other hand, there is no quality control left at all, which is generally regarded as undesirable in the field.
An alternative to this is the lottery with minimum entry requirements. This hybrid form still involves some kind of review process, but only to determine if the proposed research meets minimum requirements to be entered into the lottery system. If that requirement is
met, the research will be just as likely to receive funding as any other research that meets the minimum entry bar. This model tries to find a middle ground between a pure lottery and the current system, where minimal quality of research is preserved.
The approach that so far has attracted the most attention is the so-called peer-to-peer model. Bollen et al. (2017), in particular, proposed a model where research funding is redistributed within the scientific community, without intervention by the grant orga-nization. Effectively this means that every researcher receives a predetermined amount of funding, some fraction of which needs to be redistributed among their peers. The remainder can be kept for research as some kind of basic income. Obviously, there is a possibility that researchers receive additional funding from their peers in return. The idea behind this model is that researchers more or less know who deserves funding and who does not. Therefore it is not necessary for nwo to go around and assess individual research proposals, but it can rather just leave the decision to the researchers themselves, assuming they know what is best.
Interestingly—even though the peer-to-peer model is currently the leading candidate for the revised model—existing research pays very limited attention to how scientists would actually behave in the proposed model. They implicitly assume that researchers in the field honestly divide their funding as they value other researchers’ work. In reality, we know that people think strategically to at least some degree. A plausible strategy that directly comes to mind is colluding with a fellow academic, where you both distribute all your funding to the other to ensure more funding.
Very naturally we enter the domain of game theory, which analyzes strategic actions by players in settings such as this, called games in this context. Since research funding is of vital importance to researchers which, as we have seen, can make or break academic careers, it seems plausible that researchers (or players, in game theoretic lingo) will display strategic behavior in the game of redistributing funding. That is not to say that they do not care at all about their peers, just that their decisions are probably calculated and take into account their own interests.
In this thesis, we will define this peer-to-peer redistribution game formally in order for us to perform mathematical analysis on it. In addition to purely selfish players (more or less the standard starting point in game theory), we will consider players who have some consideration for others, which probably approaches reality better.
3 Mathematical model
We will use game theory to model the process of receiving and redistributing research funding. While in reality, this is obviously a repeated game—where decisions in the past might influence future actions—we will stick to the one-period version of the game.
3.1 The game
We model this funding process as a one-shot game, where each player obtains an individual amount of funding money, a fraction of which needs to be distributed among the other players. In turn, a player may receive funding back from the other players.
Definition 1. For a game with n players, the normal form representation ΓN specifies
for each player i a set of strategies Si (with si ∈ Si) and a payoff function ui(s1, . . . , sn)
giving the Von Neumann–Morgenstern utility levels associated with the (possibly random) outcome arising from the strategies (s1, . . . , sn). We write ΓN= [n, {Si}, {ui(·)}].
Each player i ∈ {1, . . . , n} receives an initial endowment Bi >0 at the beginning of the
game. Moreover, each player i can choose a distribution profile si = (si1, . . . , sin) ∈ [0, 1]n,
where sij represents the fraction of disposable funding that player i wishes to give to
player j. This imposes the natural restriction on the choice of strategy, namely that
Pn
j=1sij = 1 for every player i. We set sii = 0 for every player i, since players cannot
distribute funding to themselves. To summarize, we have
Si = (si1, . . . , sin) ∈ [0, 1]n: n X j=1 sij = 1 ∧ sii= 0 .
Furthermore, we introduce a fraction fi ∈ [0, 1] which is the fraction of the player’s
initial endowment that needs to be redistributed. For the time being, fi is exogenous and
therefore independent of the chosen strategy profile. In this thesis we will only consider pure strategies, as opposed to mixed strategies.
The utility function ui : S → R+ of player i is simply the monetary amount that player i
is left with after the game, that is,
ui(s1, . . . , sn) = Bi− fiBi+ n X j=1 Bjfjsji = (1 − fi)Bi+ n X j=1 Bjfjsji. (3.1)
Since sii = 0, it follows from the utility function that players have absolutely no control
over their own utility, which is relatively uncommon in game theoretic models. The players are completely dependent on the other players to improve their utility.
3.1.1 Nash equilibria
Standard practice when analyzing game theoretic models is looking for various equilibria present in the game.
Definition 2. A strategy profile s = (s1, . . . , sn) constitutes a Nash equilibrium of game
ΓN= [n, {Si}, {ui(·)}] if for every i ∈ {1, . . . , n},
ui(si, s−i) ≥ ui(s0i, s−i)
for all s0
i ∈ Si, where −i = {1, . . . , n} \ {i}, the set of all players except i.
In other words, a strategy profile constitutes a Nash equilibrium if no player can improve their utility unilaterally. Since we have already informally established that players are completely dependent on others to improve their utility, we conclude every possible strategy profile is a Nash equilibrium.
Proposition 1. Every strategy profile s= (s1, . . . , sn) ∈ S constitutes a Nash equilibrium.
Proof. Let player i ∈ {1, . . . , n} be given and consider an arbitrary strategy profile s= (s1, . . . , si, . . . , sn),
where s−i is fixed. Now consider an alternative strategy s0i for player i, so we obtain the
alternative strategy profile s0 = (s
1, . . . , s0i, . . . , sn).
given in Equation (3.1), we find that ui(si, s−i) = (1 − fi)Bi+ n X j=1 Bjfjsji = (1 − fi)Bi+ n X j=1 Bjfjs0ji = ui(s0i, s 0 −i) = ui(s0i, s−i).
Hence, ui(si, s−i) = ui(s0i, s−i). Since player i and strategy profile s0 were arbitrarily given,
ui(si, s−i) = ui(s0i, s−i) for all i ∈ {1, . . . , n} and all s0i ∈ Si and therefore s constitutes a
Nash equilibrium. Since s is arbitrarily given, every strategy profile s constitutes a Nash equilibrium.
We conclude that the standard notion of a Nash equilibrium is not very interesting in this game. However, various refinements exist.
3.1.2 Coalition-proof equilibria
Bernheim, Peleg, and Whinston (1987) have proposed the coalition-proof Nash equilibrium, which does not only allow for individuals to change their strategy, but also for coalitions to deviate from the equilibrium together.
Definition 3. Consider an n-player game ΓN = [n, {Si}, {ui(·)}]. Let J be the set of
proper subsets of {1, . . . , n}. An element J ∈ J is called a coalition. Let SJ = {Si}i∈J,
the collection of all strategy sets in the coalition. Furthermore, let −J = {1, . . . , n} \ J, the complement of J in the player set (i.e. all the players not in the coalition). Finally, for each s0
−J ∈ S−J, let Γ/s0−J denote the game induced on subgroup J by the actions s0−J
for coalition −J, i.e.,
Γ/s0
−J = [{¯ui}i∈J, {Si}i∈J}],
where ¯ui : SJ → R is given by ¯ui(sJ) = ui(sJ, s0−J) for all i ∈ J.
Now that we have a concept of coalitions in games, we can define the coalition-proof Nash equilibrium (cpne), recursively.
Definition 4.
1. In a single player game Γ, s∗ ∈ S is a coalition-proof Nash equilibrium if s∗ maximizes
u1(s).
2. Let n > 1 and assume that the coalition-proof Nash equilibrium has been defined for games with fewer than n players. Then,
a) For any game Γ with n players, s∗ ∈ S is self-enforcing if, for all J ∈ J, s∗ J is a
coalition-proof Nash equilibrium in the game Γ/s∗ −J.
b) For any game Γ with n players, s∗ ∈ S is a coalition-proof Nash equilibrium if
it is self-enforcing and if there does not exist another self-enforcing strategy vector s ∈ S such that ui(s) > ui(s∗) for all i ∈ {1, . . . , n}.
As Bernheim et al. (1987, pp. 6–7) argue, there is a potentially important restriction to the concept of cpne; only players in the coalition can deviate. That is, there is no way for a member of the coalition to ‘re-conspire’ with a player outside the coalition to further increase her utility. They can, however, re-conspire with players within the coalition. The intuition behind the coalition-proof Nash equilibrium is fairly straightforward: a strategy profile is a cpne if there does not exist a coalition J that can improve their utilities by working together, given the strategy of the non-coalition players (−J). The recursive nature of the definition provides internal consistency in the sense that in a cpne, there is also no opportunity to re-conspire within the coalition as mentioned in the previous paragraph. This consistency is achieved due to the recursive nature of the definition.
Let’s consider an example where we can show that, even though it is a Nash equilibrium, it is not coalition-proof.
Example. Consider the case where n = 3, Bi = 100 and fi = 0.5 for all i ∈ {1, 2, 3}.
That is, we have a game of 3 players where each receives an initial endowment of 100 and has to redistribute half of the funding between the other players. Let’s consider the case where every player distributes the funds evenly, to obtain a strategy profile
s∗ = ((0, 0.5, 0.5), (0.5, 0, 0.5), (0.5, 0.5, 0)). By Proposition 1 this is a Nash equilibrium.
The payoffs are ui(s) = 100 for every i ∈ {1, 2, 3} since all players receive back as much as
they give away.
However, now assume that players 1 and 2 enter into a coalition and decide to give each other all the funding, whereas player 3 plays the original Nash equilibrium. That is, we have an alternative strategy profile s0 = ((0, 1, 0), (1, 0, 0), (0.5, 0.5, 0)). In this case,
u1(s0) = u2(s0) = 125, which is strictly greater than their payoffs in the original Nash
equilibrium. Therefore, players 1 and 2 can benefit by forming a coalition together, which implies s∗ is not a coalition-proof Nash equilibrium.
1 2 3 0.5 0.5 0.5 0.5 0.5 0.5 Sharing evenly 1 2 3 1 1 0.5 0.5 0.5 0.5 With coalition
Figure 3.1: Visual representation of the example beneficial to redistribute the funding strictly within the group.
There is no known set of necessary or sufficient conditions for the existence of coalition-proof Nash equilibria in a game. (Bernheim et al., 1987, p. 7) It is easier to prove that a Nash equilibrium is not a cpne: simply provide a counterexample of a coalition that can improve its payoff by deviating. We now show that there exist no cpnes in the above game at all.
Proposition 2. For n= 3, Bi = B and fi = f for all i ∈ {1, 2, 3}, there does not exist a
coalition-proof Nash equilibrium.
Proof. Assume we have a strategy profile s∗ and assume it is coalition-proof. We can
store the strategy profiles of all players in a 3 × 3 matrix. That is,
s∗ = 0 s∗12 1 − s∗12 s∗21 0 1 − s∗21 s∗31 1 − s∗31 0 .
Note that each player only has one degree of freedom, because of the restrictions on distribution.
There are three cases: 1. s∗ 12 <1 and s ∗ 21 <1; 2. s∗ 12 = 1 and s ∗ 21 = 1; 3. s∗
12 = 1 and s∗21 <1 (wlog, we can equivalently have s∗12<1 and s∗21 = 1)
Case 1. 1 and 2 can form a coalition and improve payoff by deviating and set s12= s21= 1, since (s∗ 21+ s ∗ 31) < (1 + s ∗ 31) and (s ∗ 12+ (1 − s ∗ 31)) < (1 + (1 − s ∗ 31)),
so, case 1 cannot be a cpne.
Case 2. Assume wlog that s31<1. Then 1 and 3 can form a coalition and set s12= 0 and
s31 = 1 to collectively improve utility. So case 2 cannot be a cpne.
Case 3. The third case is equivalent to either case 1 or case 2, but then for a different set of
players, depending on the value of s∗
21 and s ∗
31. E.g., if s ∗
31 <1, case 3 reduces to
case 1, but then for players 1 and 3. By symmetry, the solutions for cases 1 and 2 apply in this case.
Since none of the cases can constitute a cpne, we reach a contradiction and conclude that
s∗ is not a coalition-proof Nash equilibrium. Since s∗ was an arbitrary strategy profile,
there do not exist any coalition-proof Nash equilibria in this case.
From Proposition 2 we conclude that there is always an opportunity to improve utility by forming a coalition. This is an undesirable property of the model, where ideally players do not form coalitions at all and simply distribute the funding as they think it should be distributed. This normative concept requires that players actually have an idea what the distribution should be in the first place.
3.2 Personal preference
To describe these personal preferences mathematically, we introduce a ‘value vector’
vi = (vi1, . . . , vin) ∈ [0, 1]n,
which represents the subjective belief of player i how the funding should be distributed. Naturally, we have the same constraint that Pn
j=1vij = 1. Note, however, that vii is not
restricted, as in si, since it is natural for players to believe that they deserve funding
themselves.
not public information. If that were the case, then the funding agency could simply gather
vi for all players i ∈ {1, . . . , n}, take the average, obtain the ‘average fair distribution’
and distribute funding accordingly. In fact, players have absolutely no incentive to make this vector public, since it could undermine their negotiation position in the game. However, this leads to games with incomplete information and the complexity that comes with it. Therefore we will consider games where all vi’s are common knowledge.
3.2.1 Fairness
So far we assumed (by construction of the utility function) that players only care about their personal monetary rewards, i.e., how much funding they personally receive. While this selfish approach is fairly standard in game theory, in reality players are probably also somewhat concerned with the fairness of the distribution. That is, they perceive a penalty if the distribution is very different from their fair value vector vi. So we can construct an
alternative utility function ˆui : S → R that takes into account this fairness payoff:
ˆui(s) = (1 − fi)Bi+ n X j=1 Bjfjsji− λi n X j=1 (µj − vij)2, where µj = P kBkfkskj P kBkfk . (3.2)
In this case, µj is the weighted average proportion that player j receives from other
players, and λi ≥ 0 is a parameter that represents how concerned player i is with the
funding of other players. The limiting case λi = 0 yields the original utility function from
Equation (3.1).
This approach has two advantages over the more selfish model: we likely gain realism, and in addition, the fairness term gives players direct influence over their payoffs. That means they generally have a unique best response to the other players’ strategy. Consider the following example.
Example. Consider the 3-player game with Bi = 100, fi = 0.5 for all i. Let λ1 = 1, and
λ2 = λ3 = 0. That is, player 1 has a nonzero fairness term, but players 2 and 3 are selfish.
Finally, let v1 = (0.2, 0.5, 0.3). In the example below, we return to players forming their
strategies without collusion—that is, we do not consider the concept of cpne for the time being.
Consider the strategy profile where player2 and 3 play the equal distribution profile. To determine the best response to the strategy profile of the other players, we maximize player 1’s utility function. Since the first two terms are independent of player 1’s strategy,
we can omit them as they are not going to influence the optimal solution. Hence, we consider max − n X j=1 (µj− v1j)2 .
This function is maximized when s1 = (0, 0.7, 0.3), which is the unique best response to
the strategies of the other players. Note that in the best response, player 1 is not playing her true value distribution, but trying to correct for behavior of the other players that she deems unfair, giving more to player2 and less to player 3.
The example introduced the concept of a best response to the strategies of other players. We found that there is a single strategy profile for player 1 where her utility is maximized, given the strategies of the other players, which corresponds to the standard definition of a best response function. We will now prove that the best response function is well-defined for players who care about fairness.
Proposition 3. The best response function of player i ∈ {1, . . . , n},
BRi(s−i) := arg max si∈Si
ˆui(si, s−i) (3.3)
is well-defined (i.e. the maximum exists and is unique) if λi >0.
Proof. Note that ˆui is bounded below by −nλi and bounded above by Pnk=1Bk (utility
can never be greater than all the money in the system). Since Si is a closed set and ˆui
is a continuous function, it attains a maximum on Si by the extreme value theorem, so
existence is proved.
To show uniqueness, note that the first two terms in Equation (3.2) are constant functions of si, hence they do not influence utility. The final term −λiPnj=1(µj − vij)2 has the
form of a negative definite quadratic function, which is strictly concave. Therefore the maximum can be attained at most once.
Since the maximum is attained at least once and at most once, it is unique and always exists. Hence the best response function is well-defined.
Given that Proposition 3 only applies if λi >0, consider the situation where λi >0 for all
i ∈ {1, . . . , n}. That means that every player i has a well-defined best response function
Proposition 4. For any player i ∈ {1, . . . , n}, BRi(s−i) solves ∂ ˆ∂sui
i(si, s−i) = 0, or it is a boundary point of Si.
Proof. Since ˆui is a quadratic (hence continuously differentiable) function, its partial
derivative ∂ ˆui
∂si exists everywhere and is continuous. That means that if BRi(s−i) is in the interior of Si, the partial derivative must be 0, otherwise it is not a maximum of ˆui. If
BRi(s−i) is not in the interior of Si, it is a boundary point of Si. Since existence of the
maximum is proved in Proposition 3, either of these cases must hold, which proves the proposition.
To see what the partial derivatives of ˆui with respect to si look like, let’s compute them
explicitly for the 3-player, constant fraction, constant endowment case as in the previous example. We use λ1 = λ2 = λ3 = 1 and v1 = (0.20, 0.50, 0.30), v2 = (0.22, 0.52, 0.26),
v3 = (0.18, 0.48, 0.34). Then the utility functions that the players must optimize are given
by ˆ u1(s) = − s 21+ s31 3 − 0.20 2 − s 12+ (1 − s31) 3 − 0.50 2 − (1 − s 12) + (1 − s21) 3 − 0.30 2 , ˆ u2(s) = − s 21+ s31 3 − 0.22 2 − s 12+ (1 − s31) 3 − 0.52 2 − (1 − s 12) + (1 − s21) 3 − 0.26 2 , ˆ u3(s) = − s 21+ s31 3 − 0.18 2 − s 12+ (1 − s31) 3 − 0.48 2 − (1 − s 12) + (1 − s21) 3 − 0.34 2 . Note that we already substitute s13 = 1 − s12 etc. in the formula since the proportions
must sum to 1. We can take the relevant partial derivatives of ˆui with respect to si to
obtain ∂ˆu1 ∂s12 (s) = −49(s12+ 0.5s21−0.5s31−0.80), (3.4) ∂ˆu2 ∂s21(s) = − 4 9(0.5s12+ s21+ 0.5s31−0.94), (3.5) ∂ˆu3 ∂s31 (s) = −4 9(−0.5s12+ 0.5s21+ s31−0.05), (3.6)
For all three partial derivatives to equal zero simultaneously, the following system of equations must hold:
1 0.5 −0.5 0.5 1 0.5 −0.5 0.5 1 s12 s21 s31 = 0.80 0.94 0.05 . (3.7)
The right-hand side of Equation (3.7) is determined by the value functions and the choice of variables, and are simply the coefficients that set the partial derivatives equal to zero. In general, we call these right hand coefficients (c1, c2, c3).
Note that the determinant of the coefficient matrix equals 0. More specifically, it has rank 2 < 3, so the matrix is singular. That implies Equation (3.7) either has no solutions or infinitely many solutions, depending on the values of (c1, c2, c3). The system has
infinitely many solutions if and only if −c1 + c2 = c3. In general, there will be no
solutions to the system. For instance, in our example, the system has no solutions, since −0.80 + 0.94 6= 0.05. From now on, we assume that −c1+ c2 6= c3, and thus the system
has no solutions.
3.2.2 Equilibria
Using the insights above, we conclude that at least one of the players must play a boundary strategy in equilibrium.
Proposition 5. In a general 3-player game, at least one of the players plays a boundary
strategy in a Nash equilibrium.
Proof. As we noted above, Equation (3.7) has no solution in general, so it is almost never
the case that the partial derivatives of the three players are all simultaneously equal to 0. However, in a Nash equilibrium, all players play a best response to the other player’s strategies. By Proposition 4, BRi(s−i) solves ∂ ˆ∂sui
i(s) = 0 or it is a boundary point. Since for any s ∈ S, at least one of the partial derivatives is unequal to 0, in a Nash equilibrium it must be the case that at least one of the players plays a boundary strategy.
A simple simulation using myopic best response dynamics, where players naively play best responses to the set of strategies played in the previous period, starting from equal distribution, confirms this finding: in this case, player 3 eventually plays s31 = 0, whereas
players 1 and 2 seem to converge to some stable distribution, which would then constitute a Nash equilibrium in the game. Please refer to Figure 3.2 for a graphical representation of the dynamics.
Every player has n − 2 free variables, because sii= 0 by definition and if the other n − 1
2 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0 Period Prop ortion s12 s21 s31
Figure 3.2: Graph of myopic best response dynamics
since every new player in the game introduces a new free variable, but also a new partial derivative restriction for each player. So larger games are still solvable using the same technique.
Since we have established that one of the players will play a boundary strategy, we can remove this player (in this case player 3) from the coefficient matrix in Equation (3.7) by deleting the corresponding row and column. The resulting 2 × 2 matrix is nonsingular, which means there is a unique distribution for the remaining two players where both partial derivatives are 0. Of course, it is not guaranteed that this strategy is on the interval [0, 1] for every player. If that is the case, then the only possible Nash equilibrium is one where every player i ∈ {1, 2} plays a strategy in the interior of Si. However, it
might be the case that the solution to the new 2 × 2 lies outside of [0, 1] for some other player. In that case, two players play a boundary strategy.
Proposition 6.1. There exists a unique Nash equilibrium in the 3-player game with
fairness (i.e. λi >0 for all i).
and compact, and each best response function BRi is continuous in si, we conclude from
Brouwer’s fixed-point theorem (Brouwer, 1911) that there exists a fixed point for the best response functions in S. This is a Nash equilibrium, since all players simultaneously play a best response in this case. So existence is proved. It remains to show it is unique. We use an inductive argument. Assume wlog that player 3 always plays a boundary strategy, that is 0 or 1, depending on her value function. Then there are two cases:
• For the remaining two players, the strategies that solve both partial derivative conditions is in the strategy set (i.e. [0, 1]). If that is the case, then that is the unique Nash equilibrium.
• If one of the equilibrium strategies falls outside the strategy set, assume wlog that player 2 also plays a boundary strategy. In that case, player 1 has a unique best response to the boundary strategies of the other players, which is the unique Nash equilibrium.
Hence we conclude that a unique Nash equilibrium exists in the 3-player game.
We obtain the notable result that, assuming that fairness is a factor in the utility of the players, there exists a unique Nash equilibrium in the 3-player case. We can now generalize this result to n players.
Proposition 6.2. The n-player game with fairness has a unique Nash equilibrium.
Proof. The proof that a Nash equilibrium exists in the n-player case directly generalizes
from the 3-player case. The same assumptions about compactness and continuity hold for
n players, hence we can invoke the fixed-point theorem again.
The uniqueness proof in Proposition 6.1 is a direct proof, based on the actual best response functions. We cannot use this technique easily in the n-player case. Instead, we use the concavity of the utility function. Since for every player i ∈ {1, . . . , n}, the utility function ˆui is strictly concave in si (cf. Proposition 3), the Nash equilibrium must be unique. This
is a well-known result, proved for instance in Rosen (1965). Hence the Nash equilibrium is unique for n players.
Even though Proposition 6.1 is essentially redundant once we have proved Proposition 6.2 (since the latter implies the former), we have a simple direct proof for the 3-player case. This is useful because it provides more insight in the actual dynamics between players in
these strategic decisions. The indirect proof of the general n-player case is less informative in that regard, hence the decision to prove the specific case first.
We can now revisit the main caveat in the original game without fairness; the lack of coalition-proof Nash equilibria. Similar to the approach used by Bernheim and Whinston (1987, p. 14) to show that a static Cournot oligopoly is coalition-proof, we can show that
the Nash equilibrium in the fairness game is in fact coalition-proof.
Proposition 7. The Nash equilibrium in the n-player fairness game is coalition-proof.
Proof. For any number of players, the Nash equilibrium is unique. That means that if we
take any coalition J ∈ J from the original game and look at the game for just the players in J, the game essentially has not changed, and the unique Nash equilibrium is still the best response to play for coalition players. Therefore, there is no enforceable deviation from the existing Nash equilibrium that the coalition J can play. In other words, there is no credible way in which any coalition of players can conspire to improve their utility. This argument can be applied to all coalitions (since the Nash equilibrium is unique for any number of players) and therefore the unique Nash equilibrium is coalition-proof. To see this, we can take the 3-player game as an example and show why it is a coalition-proof Nash equilibrium.
Example. Consider the 3-player game, and suppose s∗ is the Nash equilibrium. Now we have J=n{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}o. If we consider 1-player coalitions, the strategies of the other players are fixed. Since in the Nash equilibrium, every player maximizes his utility conditional on the strategies of the other players, this is also the case here. So for 1-player coalitions, the conditions for the cpne hold.
Now consider 2-player coalitions. We then need to look at the game that the two players in the coalition obtain, when we fix the strategy of the third player. We know from Proposition 6.2 that the 2-player game also has a unique Nash equilibrium. This equilibrium must be equal to s∗ for the players in the coalition, since the strategy of the player outside the coalition has not changed. Therefore there is no enforceable way for any coalition to deviate from the original Nash equilibrium, and thus it is coalition-proof. This reasoning generalizes to n >3 by an inductive argument. We conclude that if the
fairness parameter λi >0 for all i ∈ {1, . . . , n}, we have a unique Nash equilibrium in the
4 Discussion
The reason for developing this model is because research funding institutes are considering to implement this model for actual research funding. This proposed method is very different from the current elaborate and costly method using peer-review, which is described in detail in Section 2.1. Moreover, the proposed model has not been analyzed from a game theoretic perspective, which in this case is highly relevant, since we are invariably dealing with strategic decisions.
The main concern that instantly came up is that players might be tempted to form coalitions to improve their funding. Our analysis showed that this is indeed the case if you assume that players are only interested in their own funding. However, if players take fairness into account, this effect disappears.
It might seem that from the perspective of game theory, this model is ready to be implemented as proposed. It must be noted, however, that this analysis is no different from all other mathematical models in that it heavily relies on the assumptions. We will discuss these assumptions here.
First, we made the assumption that all value vectors are common knowledge. We did not define common knowledge in the main text (since it is a standard term in game theory), but in light of practical implications it is good to reiterate what common knowledge means. In our case, we mean that every player not only knows their own value vector, but also knows the value vector of all the other players. In addition, in common knowledge, every player also knows that all other players know each other’s value vector, and every player knows that as well. . . This infinite reasoning can be summarized as every player being completely aware that there is no hidden information about any value vectors among the players. This is a strong assumption to make in practice. We made this assumption for two reasons: to avoid uncertainty in the model (which would complicate it to a considerable extent) and to guarantee uniqueness of the equilibrium as a result.
value vector without uncertainty, let alone that they are common knowledge.
Another important assumption is that we modeled this game as a single-period decision, whereas in reality, this is obviously a repeated game, where past decisions and expectations about the future potentially play a big role in decision-making. Assuming it is public information what the distribution profiles were, it is very well possible that players are more likely to give more funding to players that gave them more in the previous period, for instance. Moreover, one can think of punishment strategies, where collusion ultimately yields higher utility than playing the single-period equilibrium.
Actually, the problem of not taking into account past observations can be accommodated in our model through the value vector. Since the value vector is exogenous, past events can be taken into account in the player’s current value of other players. The problem of not taking into account future decisions, however, is not addressed. Effectively, we just assume that the discount factor is extremely high, such that players do not care about events in the future.
Even though this is not in line with the real world, it is a well-known cognitive bias that agents pay less attention to the future than they rationally should. For instance, when faced with a problem of buying a new car today, or saving for retirement, many people are inclined to buy a car today. Hence the assumption of extremely high discount rates might not be realistic, but it is also not unreasonable to assume that this year’s funding is much more important to researchers than next year’s funding.
In addition, there is the issue of bounded rationality. We assume fully rational players in the game, even though that is not the case in the real world. For instance, in the game with positive fairness, we have shown that there is no enforceable deviation from the Nash equilibrium possible for coalitions. However, in reality, players might not be able to analyze the situation completely. That possibly leads to players entering into coalitions, even though it harms them. Especially if the game is repeated, it is likely that researchers cannot rationally analyze the game completely.
We have shown that if the researchers in the field have consistent beliefs about how funding should be distributed and they perceive a penalty if the actual distribution deviates from their fair value, this model has the potential to yield efficient results. Naturally, if necessary, further restrictions can be placed on the strategy profile. From a game theoretical point of view, the redistribution model we analyzed is capable of fairly distributing research funding in a one-shot game, if the assumptions and restrictions are met.
5 Conclusion
In this thesis we investigated the game in which players distribute research funding according to their own preferences, a game theoretical model of the system the nwo is currently considering to replace the current peer-review system.
Initially we considered ‘selfish’ games, where players are not interested in the payoffs of the other players, just their own. We found that there are infinitely many Nash equilibria in this case, since players can never unilaterally improve their payoffs. Furthermore, we proved that in the 3-player case, the cpne refinement does not hold for any strategy set. That is, there is always an opportunity for a subset of players to conspire and deviate from the equilibrium to improve their utility. This is an undesirable property of the proposed system, since ideally funding is distributed in a ‘fair’ way (regardless of the exact definition of fair), not within conspiring coalitions.
However, the assumptions of purely selfish behavior has limitations, even though it is a standard approach in game theory. It is reasonable to assume that most players incur some kind of penalty if they perceive the funding distribution as ‘unfair’. This requires that every player has a distribution of funding that they consider fair. We took this personal preference as exogenous through the value vector vi, which represents the fair funding
distribution according to player i. Furthermore, we assumed the penalty is proportional to the squared difference between actual and fair distribution for every player. This extended model is referred to as the game with fairness.
This model with fairness has many properties that are generally desirable. For instance, we proved that if every player has a positive fairness parameter λi, there exists a unique
Nash equilibrium in the game, and subsequently that this Nash equilibrium is coalition-proof. Obviously, the characteristics of the Nash equilibrium are heavily dependent on the exact values of the value vectors on the one hand, but also on the magnitude of the
payoff, but also in the payoff of others.
The most important result is that the fairness model is coalition-proof. The implication that it is impossible to successfully form coalitions if all players care about fairness to any
degree—the only restriction is λi >0 for all i, magnitude is irrelevant—renders this model
a much better candidate for actual implementation as a research funding distribution model. It eliminates the concern that researchers will conspire for personal gain, which was one of the main arguments for analyzing this system game theoretically in the first place.
In future research, it could be tested whether the mathematical results in this thesis hold in an actual experiment. An experimental approach could reveal if the assumptions in this model hold in reality, and if actual strategies correspond to the predictions in this thesis. In addition, the repeated nature of the model (which is not addressed here) is possibly easier to analyze in an experimental set-up than in a purely mathematical environment, since the bounded rationality constraints that are present in reality, are probably much more pronounced when considering the repeated game. It would be interesting to observe how actual subjects redistribute the research funding in an experimental setting, providing further insight in the actual attractiveness of this peer-to-peer model as an alternative to the current system.
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