Evolution of Government's Performance Through Yardstick Competition in the Spatial Game of Gubernatorial Elections

Yardstick Competition is a unique feature of gubernatorial elections and may have a paramount role in the development of democracy and local governmentâ€™s performance. This paper investigates the behaviours of governors and voters in an evolutionary game of gubernatorial election by introducing the spatial simulation process where voters can make comparisons between the incumbent and neighboring politicians. Based on the model, we portray the evolutionary process and topologies of local governmentsâ€™ performances in federal systems. In the baseline model, we find that the variance of governor candidatesâ€™ performances, as well as the intensity of the yardstick competition, are positively related to the overall performance of governments. To study the relationship between elections and foreign policies, we employ an evolutionary Public Good Game in which governors can affect the welfare of neighbours by determining whether to invest in cross-provincial constructions. In the extended simulations where governors and voters are assigned to various characters, we find that asymmetry between candidatesâ€™ potentials and votersâ€™ perception increases the uncertainty of the electoral results, and selfless voters will lead to lower performances of governments.


Introduction
. A federal system usually has several layers of local governments that generally mirror the federal government.
The election of local governors, compared to the presidential election, is closer to the citizens' daily lives but has raised less attention from scholars. Researchers have noted the uniqueness of the gubernatorial elections compared to the senatorial or presidential elections in a federation (Stein ; Atkeson & Partin ): voting for local governors, as state executives, highly depends on the perceived local economic conditions, whereas senators' fortunes are linked to the president (Atkeson & Partin ). Empirical studies have confirmed a positive correlation between the local economy and promotion or reelection probabilities of governors, whether in real federal systems such as the United States (Peltzman ) or in quasi-federal states such as China (Tsai ; Li & Zhou ).
. Yardstick competition is another unique feature of gubernatorial elections. A burgeoning literature of political studies has revealed the existence of yardstick competitions between jurisdictions (Breton ; Costa-Font et al. ) at country level (Devereux et al. ), state level (Wheaton ; Chirinko & Wilson ) and municipality level (Brülhart & Jametti ; Bordignon et al. ). Originating from Shleifer ( ), the model of yardstick competition applied to political economy was formally developed by Salmon ( ), followed by Besley & Case ( ). Political yardstick competition emerges when the governmental performance in various jurisdictions becomes su iciently comparable. Voters can observe the performance and adopt it as an instrument to evaluate their governments (Besley & Smart ; Allers ). This alleviates the agency problem by making meaningful comparisons between jurisdictions (Bodenstein & Ursprung ). .
There are two types of yardstick competition applied in gubernatorial elections. The first type is the so-called 'yardstick competition to be promoted', where a successful incumbent o icer in a lower-level jurisdiction may Model Baseline model Setup . We first consider a L×L = N lattice representing the federation. Each site on the lattice is occupied by a lowerlevel jurisdiction called province and headed by a governor. Note that unlike most network approaches, the lattice here shall not meet the periodic boundary conditions since boundary provinces exist in the real world. Thus, in the baseline model, each province in the federation is geographically identical except for provinces that share boundaries with the federation. The performance of the governor is measured by b i (t), where i(1 ≤ i ≤ N ) denotes the index of province and t denotes the term of o ice.
. Initially, governors are randomly appointed to provinces by the federal government. The variable b i (0) ≥ 0, which is the performance of the first governors, takes a value from to randomly (this initial setting will be relaxed in later sections). The mutation that changes the value of b i in steps of ±0.01 occurs at a constant mutation rate µ.
. At the end of each term of o ice t, the gubernatorial election between the incumbent and a challenger takes place. The settings of the gubernatorial election are partly adopted from Bodenstein & Ursprung ( ). The challenger has no record as a political elite, and thus voters cannot estimate the challenger's performance bc i (t), if in o ice, according to her past performance. Simply but without loss of generality, we assume that the voter's expected bc i , E(bc i ), would be the average b value of this province's nearest neighbouring incumbent governors (the von Neumann neighbourhood without the focal governor): The parameter n i denotes the number of neighbours for province i. Note that some provinces sharing boundaries with the federation may have only or neighbouring provinces, so n i takes a value from to . M i is the set of all neighbours' indexes of province i.
. Under this circumstance, the probability that the incumbent governor of province i wins the reelection held at the end of term t is: The Expression is taken from Tullock ( ). The parameter φ ≥ 1 measures the extent of the yardstick competition between the incumbent and the challenger. If φ = 1, the yardstick competition is complete and no incumbency advantage is imposed in this election.
. Once the incumbent is reelected, she will run the next term in o ice with the same b i (but remember that the mutation may occur at the beginning of each term). If the challenger wins the election, she will be the next governor until the next election. She runs her governorship with bc i if no mutation occurs. In Sections . -. , we will set di erent types of distributions of bc i . Once the incumbent governor loses the election, she will never become involved in any subsequent elections. In addition, there is no life limitation for governors; that is, if elected in every election, the governor can be in o ice forever. .
The model settings above imitate the typical yardstick competition in politics (Shleifer ; Besley & Case ) and we are interested in the simulation outcome for such model settings in a cellular game. The algorithm for the baseline model is given in Algorithm . We envisage the following questions according to the simulation results: . Will democracy and yardstick competition lead to better performances of local governments in general?
. What spatial characteristics of government performances will emerge given di erent initial settings?

Algorithm Algorithm for the Baseline Model
Require: Federation size L, mutation rate µ, competition parameter φ, Number of generations T . Initial performance for each governor i: b i ,i = 1, 2, ..., L × L. Place each governor on one of the cell of the lattice. : while iteration smaller than T do : for each governor i do mutation algorithm : Sample n from uniform distribution U (0, 1) : if n < µ then p i : the probability of the incumbent to be reelected : Sample m from uniform distribution U(0, 1) Begin the election : if m < p i then when the incumbent wins the election : Reelection matrix[province i's coordinate]← 1 Reelection matrix is used to mark down the spatial distribution of election result. If an entry in the matrix is marked , then the corresponding governor is reelected. : else when the challenger wins the election : Reelection matrix[province i's coordinate]← 0 : Sample bc i from a specific distribution D with mean E(bc i ) In the paper, D is the uniform The challenger is now in the o ice

Simulation results
Under di erent bc i distribution .
As discussed in the previous section, the distribution of bc i may take various forms. We propose di erent types of the distribution and then test the outcomes under these bc i distributions. .
We generally consider uniform distributions in the following subsection. All the distributions that bc i follows should meet the basic requirements that the mean of the distribution should be E(bc i ), if we assume that voters' estimation of the challenger's performance is appropriate. The assumption holds for rational voters in the long term: if the distribution's mean is larger or smaller than E(bc i ), voters will eventually notice this discrepancy and then adjust their estimation of bc i to the mean value. .
The main result of the simulations in Figure is that the average b value across the federation (i.e., the federal average performance of governors) will increase over terms in o ice under these bc i distributions. In particular, the value of α, representing the potential range of the distribution, is positively related to the average b value in general. A larger range of bc i , namely the value of α, will provide voters with a larger range of the challenger's possible performance. Due to the voting system and yardstick competition, the newly elected governor with larger bc i will enjoy support from voters, and the newly elected governor with lower bc i will lose her governorship soon a er, so despite the risk of electing an incompetent governor, voters will benefit from the uncertainty of the challenger's performance in the long run.
(a) (b) Figure : Average b values in the federation among generations. All average b values increase compared to the initial settings. The e ect of di erent distributions of bc i is significant in the development of the average b value. In subplots (a) and (b), bc i follows the uniform distribution U (αE(bc i ), (2 − α)E(bc i )). The values of α are listed in the legend. Other parameters include L = 10, µ = 0.01, φ = 1. Each data point in the plot is averaged over realizations of simulations. .

Result :
In a certain range, the larger the range of the challenger's possible performance, the better the overall performance of the governors in the federation. .
It is worth noting that in Figure , the time length we used in the simulation is terms (and it would be even larger in the rest of the paper), which is equivalent to ∼ years in reality. Obviously, a continuous year election has never appeared in history, but we run the simulation for at least steps to ensure the robustness of the result and show more details for the evolutionary paths.
Under di erent initial distributions of b i .
One of the major advantages of the cellular automata is to enable readers to observe the spatial characteristics directly. We are interested in the spatial distribution of governor's performance in the model of gubernatorial election under di erent initial settings.
. Figure shows the simulation outcomes in the th term (right column) and the corresponding initial distributions (le column). It is observed from the outcomes that high-level-and low-level-performance provinces are clustered in di erent regions in the panel, resulting in the gradient of governor's performance across the federation. The initial settings (le column of Figure ) are randomly generated to describe the situation where each governor was initially given a random performance b i ∈ [0, 1]. By comparing the initial distribution and the simulation outcome, we cannot assert any links between the initial distribution of governors' performances and the outcome; thus, we believe that the random fluctuation creates the gradients of governor performances from the initial randomness.  Figure . .
The basic intuition beneath the phenomena of performance gradient follows the key idea in yardstick competitions: a governor may lose incentives to improve her administrative ability or to embark on reforms if her neighbouring provinces are also su ering from their governors' bad performances since the voters may downgrade their expected ratings on the incumbents according to the decrease in the welfare of the neighbouring provinces. In the same fashion, a governor whose neighboring provinces exhibit better development level faces great pressure to improve his performance in order to survive the next election. .
The setting bc i = E(bc i ) indicates that there is no uncertainty about the challenger's performance, so voters are aware that the next governor, if the challenger is elected, will be of the average level of performance of their neighbouring governors. The lack of uncertainty drives the performance distribution into high agglomeration. Once the uncertainty of the challenger's performance emerges, the spatial characteristic of agglomeration disappears and turns into disordered patterns gradually because of the increasing uncertainty (see Figure ). The value of α indicates the range of performances of the challengers. .
To compare with the spatial outcomes in Figure  .
In Figure , the initial distribution of governors' performance represents the division of the federation between the well-governed provinces on the lower le half and the provinces that are su ering from poor governances on the upper right half. In Figure , few well-governed provinces are surrounded by the poorly-governed majority, and the distribution goes opposite in Figure . It is shown in Figure -that the initial topology of the distribution will no longer survive a er enough long time steps. Instead, the heterogeneity in governors' performances may increase over time. For example, in Figure , the initial di erence in performance between the best and worst governor is . , and the di erence climbs to more than . in the th term, and more than . in the th term. To conclude, the gubernatorial election improves every governor's performance, but is also likely to widen the gap between provinces. .

Result :
The initial gradient or clusters of governors' performances will vanish over time due to gubernatorial elections and yardstick competitions, although the heterogeneity in governors' performances may increase. .
In this part, a constraint is made that if b < 0.01, no mutations will occur to ensure b ≥ 0. Due to this constraint, if, initially, every governor's b equals , this low-level equilibrium will never be broken. To test the role of mutation, the constraint is relaxed such that mutations can occur for any value of b and b < 0 is acceptable. The simulation results under the relaxed condition are shown in Figure   . Figure shows that even at a very low mutation rate (µ = 0.01), fluctuations will break the low-level equilibrium. Even though the average b value was finally less than , the federation enjoys comparatively high average b values before the th term. .
Similarly, the process of the evolution of b i when, initially, every governor obtained is also based on mutations.
The simulation result is shown in Figure

Robustness check .
Robustness checks on various parameters (L, µ and φ) and iterations (terms in o ice) are performed to ensure the power of the model. The simulation results in previous sections are robust with respect to the initial settings. Several meaningful test results are reported in this subsection.
. It is well known in the field of innovation economics that a larger number of lower-level jurisdictions will result in a higher overall performance of policies in a federal system when policy innovation and intimation are conducted by jurisdictions (Linge ; Saam & Kerber ). To investigate the role of the number of provinces (N = L * L) in the election, we test the hypothesis in our gubernatorial model by performing a robustness check on the parameter L. .
In contrast to the classical outcome in innovation economics, Figure shows that the number of provinces is not an important factor involving the evolution; that is, we cannot predict the comparative outcomes of average performance, ceteris paribus, from the numbers of provinces. The absence of the higher-number-of-provinces advantage in the gubernatorial game may result from the voter's bounded rationality in information: one can only observe the b values of one's neighbours, and thus other members of the federation cannot a ect the election directly. .
The value of φ measures the intensity of yardstick competitions at the province level (Bodenstein & Ursprung ), where φ = 1 denotes the complete yardstick competition, and a higher φ denotes an incomplete competition with an incumbency advantage between governors. .
Figure shows the simulation of models with di erent φ. The more intensive the yardstick competition (i.e., smaller φ), the higher the long-term overall performance of all governors. When φ = 100, where the e ect of yardstick competition is small enough, the average b value increases slightly during the terms. It is possible to predict the extreme outcome in which the overall performance undergoes no change if φ goes to infinity. The simulation result signifies the importance of the conditions a ecting the value of φ including incumbency advantage, free flow of information across provinces, and the comparability of the performances between governors. Figure : Average b values in the federation with di erent φ values. Other parameters include α = 0.6, µ = 0.01, φ = 1, term= , . Each data point in the plot is averaged over realizations of simulations. .
The robustness check with respect to the mutation rate µ implies that mutation does not significantly a ect the overall performance ( Figure ), except for the situations in Figures and . .

Result :
The number of provinces and the value of the mutation rate are not the main contributors to the evolution of the overall performance in a federation. However, mutations may be the driving force to break the low-level equilibrium.
Further discussion: Geometric random walk model and simulation length .
In the previous subsections, we have described the dynamics of the average performances of governors by figures. The topology in these figures suggests that there is a resemblance of average performance histories to a geometric random walk. The model of random walk has been widely used in financial time series and other social sciences to describe and forecast the related process (Voit ; Barrat et al. ). In the geometric random walk, the natural logarithm of the process is assumed to walk a random walk (Nau ). To illustrate the relations between the evolutionary path of government performances and geometric random walk, we plot the natural logarithm of the average b value from a typical simulation result of periods.
. Figure , just like any other previous plots, presents a similar pattern of exponential growth with occasional oscillations. The log plot of average b values, in a rough term, resembles a linear trend fitting. In Figure , the first di erence of logged series can tell us more about the statistical feature of the process. It is easy to tell that there is a dri di erent from zero in the di -log plot. Actually, the mean change is . , while the standard deviation is . , which means that from one term to the next, the index grows on average by . %. .

The average b value plot in
The pure noise pattern of the di -log plot indicates that the geometric random walk model is a simple but efficient way to model the process (for a discrete geometric random walk {X t }, its di -log log X t+1 − log X t for all t is a white noise ξ by definition). In particular, the evolution of government's performance can be modeled as: with a symmetric, zero-mean jump distribution Ψ(ξ), and c is the constant dri . In the simulation in this subsection, c = . and the standard deviation of Ψ(ξ) is .
The memoryless property of the geometric random walk model can help us simplify the simulation. The property illustrates that the changes in natural logarithm from one term to the next is not related to the history of the process, which permits the expected growth in average b for all the terms followed by. Therefore, the average performance of governments will continue to grow, although some short-term volatility may be observed. The memoryless property can simplify our following practice: we no longer need to perform too many simulation steps to permit the robustness of the results.

Models with inter-jurisdictional a airs
Setup .
The impact of foreign countries' policies and incumbent's foreign policies on the presidential election has enjoyed lasting interest among scholars (Risse-Kappen ; Smith ; Small ; Mead ), but few researchers have focused on the relations between inter-jurisdictional policies and gubernatorial elections. Regardless of the home a airs such as local tax policies, lower-level jurisdictions, if enjoying far-reaching autonomy as the German Bundesländer or the Belgian regions, are also responsible for dealing with foreign a airs with their neighbours and other members of the federation. As pointed out by Halperin et al. ( ), foreign issues definitely a ect the presidential election in the United States. The inter-jurisdictional policy (executed inside the federation) not only a ects the welfare of the jurisdiction but also plays an exogenous role in the calculation of other member states' welfare, especially for the jurisdiction's neighbours. Obviously, jurisdictions will receive more policy externality impacts from their neighbours than other member states, resulting from the externality of public goods, whether natural (e.g., water resource sharing by jurisdictions) or artificial (e.g., a hospital also serving the neighbouring citizens), and direct interactions such as trade, aid or even conflicts. .
As sovereign entities within the federal system, in our model, each province has its own authority to deal with inter-jurisdictional a airs, although the authority may be restricted and supervised by the federal government.
In this section, we model the inter-jurisdictional a airs with a conventional evolutionary public goods game (PGG) involving two neighbouring governors as players. One good example is the construction of cross-provincial public infrastructure, which is beneficial to both provinces. We assume that the governor can play either 'Contribute (C)' or 'Free ride (F)' as her foreign strategy. The payo s for the contributor and free rider are: where c is the contribution cost (c > 0), r is the multiplier of return (r > 1) due to economies of scale, and N c denotes the number of governor(s) playing C in the game (N C ∈ {0, 1, 2}, N C + N F = 2). The settings of the PGG are adopted from Ye et al. ( ). Then the foreign performance can be written as: in which the variable Payo ij represents the payo to the province i from the PGG between Province i and j. .
For simplicity, in this section, we suppose that the governor will be consistent in choosing the strategy: a er choosing C or F to deal with foreign a airs, the governor will play this strategy with each of his neighbours and will not change the strategy as long as the governor is in o ice. This constraint will be relaxed in later sections. .
The payo from inter-jurisdictional a airs is also considered to be a composition of the governor's performance; where bh i is the governor's performance in home a airs, which is the same as the b i in the baseline model. The parameter bf i is the governor's performance in foreign a airs (inter-jurisdictional a airs), which is the same as the payo for the governor in the PGG. The parameter θ(0 < θ < 1) measures the weight of home a airs in evaluating the governor's performance. Under these circumstances, the gubernatorial election still follows Equation in the baseline model.

.
The challenger's performance in inter-jurisdictional a airs, if elected, will be more complicated than in home a airs as defined in the baseline model. First, we assume that the voters cannot disengage bf i from b i ; the voters can only observe the overall performance of the governor. Since the voters cannot monitor bf i directly, they are indi erent to foreign strategies and have no moral preference on either strategy. In this case, the challenger is free to choose her C or F in the PGG. For the dynamics of the foreign strategies, we assume that the elected challenger (whose term in o ice is t) is politically rational, and she makes the inter-jurisdictional decision according to her neighbouring provinces' strategies in term t − 1 to maximize bf i (t), although the neighbouring provinces' strategies may alter in term t. Pre-tests indicate that more attention should be paid to the behaviors of governors when they are indi erent to playing C and F (i.e. Playing C and F will result in the same payo ). If we force the governor to accept F when she is indi erent to both strategies, then the equilibrium would be that every governor plays F. Thus, we assume that when the governor is indi erent between C and F then she will randomly choose C or F with equal probabilities.
. The probability of a challenger obtaining the governorship is the same as Equations and , and bc i ∼ U (αE(bc i ), (2 − α)E(bc i )) ( ) states the realization of challenger's overall performance once elected. Algorithm shows the algorithm for the model with inter-jurisdictional a airs.

Algorithm Algorithm for the Model with Inter-Jurisdictional A airs
Require: Federation size L, mutation rate µ, competition parameter φ, Number of generations T , PGG parameters R, S, T, P , weight parameter θ. Initial domestic performance bh i , and the foreign strategy f s i ∈ (cooperation, f reerider) for each governor i, i = 1, 2, ..., L × L. Place each governor on one of the cells on the lattice. : while iteration smaller than T do : Calculate N C and N F N C : number of cooperative governors, N F : number of defective governors : for each governor i do : Sample n from uniform distribution U(0, 1) : if n < µ then for each governor i do begin PGG : if f s i =cooperation then : for each i's neighbor governor j do : if f s j =cooperation then : bf if f s i =f reerider then : for each i's neighbor governor j do : if f s j =cooperation then : bf Sample bhc i from a specific distribution D with mean E(bhc i ) : bh
In this subsection we continue to study the voters who are set to care about their own provinces. We call them selfish voters, whose idea can be traced back to the basic utility hypothesis for the calculus of voting (Riker & Ordeshook ). For selfish voters, their utility of voting is independent of other provinces, suggesting that their voting behaviour is determined by their governor's performance only. The selfish voters are designed to behave in the same manner as the voters described in the baseline model, and the only di erence is that the selfish voters in this particular subsection consider the entire performance (i.e. b i (t) = θbh i (t) + (1 − θ)bf i (t)) of the governors and challengers, rather than the performance of home a airs alone. .
Figure depicts the evolutions of both overall performances of governors. Under these initial settings and update formulae, all governors will choose the strategies of F, and thus bf i = 0 for i = 1, 2, . . . , N , which is a common result of the public goods dilemma (Olson ; Hardin ; Dionisio & Gordo ). In addition, the evolutionary path of the performance in home a airs bh i is similar to the path of b i in the baseline model (see Figure ). To be particular, for Figure (b), the dri of the geometric random walk model is . . Since all governors play F infinitely, the game with inter-jurisdictional a airs does not di er from the baseline model, and thus the evolution of the performance in home a airs follows the update dynamics in models without foreign a airs.
(a) (b) Figure : Overall performance and strategies for governors in and , terms. The labels of 'cooperative provinces' and 'defective provinces' denote the number of governors using the strategy of C and F, respectively. The distribution of bhc i follows U (0.6E(bc i ), 1.4E(bc i )) and, initially, bch i takes a value from to . We initially set governors to play C and governors to play F, and the distributions of the C-and F-playing governors are random. Other parameters include φ = 1, µ = 0.01, θ = 0.5, c = 1, r = 1.5. The simulation result is robust with respect to time. .
Since each inter-jurisdictional game only involves players, we can simply conclude the payo s in the following payo matrix: In this simulation, we can compute the payo matrix with the parameters given in Figure : R = 0.5, S = −0.25, T = 0.75, P = 0. Note that the payo of the free rider in the unilateral contribution, T , is larger than the payo for mutual contribution, R. Intuitively, we test the simulation result for T < R, when the best payo for a free rider (T ) is less than the payo for a contributor when mutual contributions occur (R). Figure : Overall performance and strategies for governors in , , terms. The distribution of bhc i follows U (0.6E(bc i ), 1.4E(bc i )) and, initially, bch i takes a value from to . We initially set governors to play C and governors to play F, and the distributions of the C-and F-playing governors are random. Other parameters include φ = 1, µ = 0.01, θ = 0.5, c = 1, r = 3. The simulation result is robust with respect to time.

.
Figure shows the reversal of strategies when R = 2, S = 0.5, T = 1.5, P = 0. The federation has reached an equilibrium where every governor plays C in every term (not shown due to the limitation of length). Meanwhile, the overall performance in home a airs reaches the low-level equilibrium in the first terms compared with the simulation in Figure . Simulation (a) b distribution (top right of Figure ) portrays the diamond-like spatial distribution of the performance b i , which is due to the lack of the periodic boundary condition (see Sections . -. ). Fewer yardstick competitions for a governor will lead to lower performances in borderlands. When R > T , governors have no incentive to play F anymore; thus more neighbours means more inter-jurisdictional games and more payo s.

.
We find that given enough long terms the performance in home a airs will experience unexpected growth a er certain periods (the turning point in Figure (b) is terms). Comparing Figure and , the late development of governor's performance when R < T cannot catch up with the development when R > T . Note that in the game with inter-jurisdictional a airs, the voters cannot disengage bf i from b i ; the voters evaluate the challenger's performance according to the average value of the neighbouring governors' overall performance b i , rather than the performance in home a airs only (bh i ). When R > T , all governors play C, and thus bf i reaches the maximal value for each i. In this circumstance, the gap between incumbents with low and high bh i is relatively reduced because the common high-level bf i has compensated part of the di erence between governors. Consider an extreme case when bf i = ∞; then the voting systems lose all the incentive to promote bh i since the governor's performance in home a airs is completely ignored. Namely, in the first terms, the low-level equilibrium trap for performances in home a airs can be ascribed to the emergence of the high-level equilibrium for performances in foreign a airs. The payo s from inter-jurisdictional games are large enough for the voters to ignore the performance inside the province, and therefore the evolution of bh i stops. To verify this assumption, we make the following robustness checks with respect to the value of θ, the weight of home a airs when evaluating the governors in Figure . (a) (f) (g) Figure : Overall performance for governors in and , terms with di erent values of θ = . , . , . , . for the four subplots, respectively. The distribution of bhc i follows U (0.6E(bc i ), 1.4E(bc i )) and, initially, bch i takes a value from to . We initially set governors to play C and governors to play F, and the distributions of the C-and F-playing governors are random. Other parameters include φ = 1, µ = 0.01, c = 1, r = 3. .
The simulations in Figure indicate that a lower weight of inter-jurisdictional a airs in evaluating the governor will lead to better governor performances in home a airs. This conclusion is widely applied in politics when public o icials try to create foreign emergencies to divert voters' attention from domestic a airs. However, this low-level equilibrium will be eventually broken in the future even if the weight of foreign a airs is large. The breakthrough may be ascribed to the accumulation of time, which eventually induce some good-performance governors, and once the average bh i breaks the threshold created by foreign a airs, nothing can act as a brake on growth. This process may explain the exponential growth of bh i in some subplots of Figure . .
Once voters observe bh i and bf i separately, the foreign a airs no longer influence the governor's performance in home a airs, and the simulation results will be similar to the simulation results in Sections . -. .

Result :
A lower weight of inter-jurisdictional a airs in evaluating the governor will lead to the governor's better performance in home a airs when all governors are free riders.

Selfless voters .
In the previous sections, we took it for granted that all voters were rational and selfish, and their behaviours were entirely based on the welfare of their own provinces, taking neighbouring provinces' welfares as references. Additionally, in most studies on yardstick competitions or gubernatorial elections, the jurisdiction is isolated from the federation, focusing on a single governor's political performance but not on the spatial characteristics covering other jurisdictions. Moreover, the behaviours of voters in previous studies were the univariate functions of an individual's own utility or the local social welfare (McNutt ). This assumption is taken for granted in presidential electoral studies, indicating that voters care about foreigners only if foreigners can influence their own country (e.g., via invasion, annexation). However, at the level of gubernatorial elections, local voters o en have social preferences (Ho man et al. ) which may overflow to neighbouring jurisdictions due to the recognition of national identity in the same federation or the close relations between voters and neighbouring citizens. In the research field of cooperation, scholars have found that taking neighbours' payo s into consideration could be the driving force for the emergence of cooperative behaviours in network games (Chen et al. ; Li et al. ). It is appropriate to transplant the altruistic characteristics for players from the evolutionary cooperation literature to the voters as one of the alternatives for voter's type. .
We first assume that all voters are of bounded selflessness; namely, they only care about the von Neumann neighbourhood. The adjective 'bounded' here means that the local voters show no interest in the welfare of provinces that share no boundaries with their own province of residence. The assumption of boundless selflessness is only appropriate for the presidential election, and therefore bounded selflessness is a remarkable feature of gubernatorial elections. .
Following the tradition of philosophy, we consider the following two types of selfless voters: The Utilitarian voter (Bentham ) and the Rawlsian voter (Rawls ). Utilitarian voters evaluate the governor according to the summation of the neighbourhood's (including their own province) performances in both home and foreign a airs: where the superscript U denotes the voter's type of Utilitarian. Q i is the set of all provinces in the von Neumann neighbourhood of province i. Rawlsian voters judge the governor by the minimum of all provinces' welfare in the neighbourhood (including their own province): where R denotes the Rawlsian voters. .
Although the selfless voters predict the challenger's potential using the average value of neighbouring provinces' W U i or W R i , the actual distribution of challenger's performances in home a airs still follows the distribution U (αE(bhc i ), (2 − α)E(bhc i )), as in the previous sections. Since the voters cannot distinguish the performance in home a airs from foreign a airs, they can only evaluate the challenger using the average performance in both a airs of all neighbouring governors, but it is only accurate to estimate the challenger's performance in home a airs according to its neighbours because the challengers will choose foreign policies based on the policies of the incumbents nearby in the last term. Therefore, the existence of foreign a airs will induce the information asymmetry in voters as side e ects even if θ = 0. .
Equations and have the implication of justice, which means the voters are more sensitive to the welfare of neighbouring citizens. This assumption of justice is based on the fact that the governor's strategy in interjurisdictional a airs can a ect the welfare of neighbouring provinces; thus, the voters should also be responsible for their neighbours' welfare when electing their own governor. However, the local governor cannot a ect the welfare of non-neighbour provinces, so local voters will ignore the non-neighbour citizens. .
The dynamics of the election mirror Sections . -. ; only the expression for E(bc i ) changes for di erent types of selfless voters. For Utilitarian voters: and for Rawlsian voters: Note that the voters now no longer evaluate governors and challengers by their own (expected) performance, but vote according to the expected (Utilitarian/ Rawlsian) welfare of the neighbourhood via Equations -.
In simple words, the updating formula for election does not change, except the expression for E(bc i ), which is now re-defined by Equations and , and thus the re-election probability (Equation ) is now: . Figure (a) results in the situation in which all governors play F when R < T . Unexpectedly, except for the public goods dilemma that emerged in the elections with selfish voters, the Utilitarian voters have created a low-level equilibrium for average performance. Although the patterns for (a) and (b) look di erent, both simulations in (a) and (b) can be described by geometric random walk models with negative dri s (bh(t) = e x(t) , x(t) = x(t − 1) + ξ + c). For R < T , c =-. and for R > T , c =-. . The dri s tell us that regardless of the relations between R and T , the average performance will decrease continuously and slowly in foreseeable future. The major di erence in (a)   terms with Utilitarian voters. The distribution of bhc i follows U (0.6E(bc i ), 1.4E(bc i )) and, initially, bch i takes a value from to . We randomly set governors to play C and governors to play F, and the distributions of the C-and F-playing governors are random in the first term. Other parameters for simulations in (a): φ = 1, µ = 0.01, c = 1, r = 1.5, θ = 0.5. Other parameters for simulations in (b): φ = 1, µ = 0.01, c = 1, r = 3, θ = 0.5. The dri in (a) is -.

Conducting simulation models of elections with Utilitarian voters in
, and in (b) is -. . Figure : Performances for governors in home a airs with Rawlsian voters. Initially, bch i takes a value from to . We randomly set governors to play C and governors to play F, and the distributions of the C-and F-playing governors are random in the first term. The distribution of bch i follows U (0.6E(bc i ), 1.4E(bc i )), E(bc i ) = 1 ni j∈Mi b j . Other parameters include φ = 1, µ = 0.01 , c = 1 ,θ = 0.5. For R > T , r = 1.5; for R < T , r = 3. Each data point in the plot is averaged over realizations of simulations. .

Figure
shows the results for Rawlsian voters. When R < T , although all the governors' performances in foreign a airs are zero, the performances in home a airs exhibit a surprisingly exponential growth. When R > T , the simulation results are not consistent: most cases report the flat evolutionary paths as in Figure (b), but di erent types oscillations may occur randomly over time. Compared to the outcomes of the Utilitarian voters, the overall performances are significantly promoted when R > T but reduced when R < T in a federation with Rawlsian voters. Figure : Overall performances for governors in , terms with Rawlsian voters. Initially, bch i takes a value from to . We randomly set governors to play C and governors to play F, and the distributions of the C-and F-playing governors are random in the first term. The distribution of bch i follows U (0.6E(bc i ), 1.4E(bc i )), E(bc i ) = 1 ni j∈Mi b j . Other parameters include φ = 1, µ = 0.01 , c = 1 ,θ = 0.5; for R < T , r = 3. .
Result : Altruistic voters will lead to lower performances of governments regardless of the types of altruistic voters (Utilitarian or Rawlsian) and the relations between R and T.
. Utilitarian voters evaluate challengers by the welfare of neighbours' neighbours, and Rawlsian voters vote according to the local minimal performance: the concern of selfless voters is far from the welfare of their own province. Thus voters no longer stand for the interest of the local citizens and governors no longer work for local welfare. This results in the tragedy of kindness as stated in Result .

.
For the Rawlsian voters, consider a province with local minimal performance; according to Equations , and , we have and thus the probability of governor's reelection will be: . Thus, given enough terms, the governor is highly likely to fail the election. Since the challenger's performance in home a airs follows the uniform distribution ranging from 0.6E(bhc k ) to 1.4E(bhc k ) where E(bhc k ) > bh k , once the challenger is reelected, the province k will enjoy better performance with relatively high probability. Similarly, if province z is one of k's neighbours, we still have P z = φ φ+1 regardless of b z . This situation causes many oscillations of the evolutionary paths.

Conclusion and Discussion
. Although not uncontroversial, the application of cellular automata simulation in electoral studies captures many aspects of democracy at the gubernatorial level. In contrast to the traditional orientation of political studies, the evolutionary and spatial studies on elections in this paper place much weight on the process and transformation of the social welfare, which is expressed as the governor's performance driven by the voting system. The foregoing exercises of simulations have illuminated several problems concerning the evolution of governors' political performances in provinces.
. Both yardstick competition and politics have been and remain today popular subjects among scholars, but most papers involving in these two topics focused on tax setting/mimicking (e.g. Besley & Case ; Bordignon et al. ; Allers & Elhorst ) and provision of public services (e.g. Revelli ; Terra & Mattos ), rather than the basic intuition beneath the election outcomes, and its relations with both voter's behaviours and crossborder interactions. Although rooted in the context of Tullock ( ) and the Bodenstein-Ursprung ( ) models for fiscal choice/ economic behaviours, this paper o ers proofs of some intuitive statements in the gubernatorial elections in the environment of a simple spatial gubernatorial game. .
The main purpose of this paper is to suggest a method of computer simulation to model the dynamics of electoral outcomes. Following the previous observations, we propose the following results concerning the governors' performances. From the side of governors, the following three findings are worth noting: (i) larger range of the challenger's performances will induce better overall performance of governors in the entire federation; (ii) there is no evidence supporting that the initial distribution of governors will a ect the overall and specific states' performance in the long run; and (iii) the mutation of governors' performance drives the entire federation out of a stable equilibrium, and a fiercer competition among the governors will lead to a better overall performance. From the side of the voters, the first finding is that a lower weight from the voters evaluating the inter-jurisdictional a airs induces better governor performance when all governors are free riders. Moreover, we find that altruistic voters will trap the performance into a lower level. .
Although some conclusions are commonly observed in political practices, others seem to be meaningful only with highly strict assumptions. Keeping to these conclusions, we may conjecture the possible potential of democratic systems. More precisely, we underline the importance of voters' perception of both the incumbent and the challenger in the evolution of democracy. The cardinal di erence between selfish voters, Utilitarian voters and Rawlsian voters is not in their moral virtues; the di erence lies in the methods of evaluating the incumbent and predicting the challenger. Notwithstanding that we have analysed three types of voters, the most e icient method to promote politicians' performance is to ensure the information transparency of the candidates for the governorship or presidency. In this paper, the information asymmetry or the lack of information transparency is represented by the mismatch between challenger's performances and voters' perceptions and the impossibility of distinguishing incumbents' performances in home and foreign a airs. The asymmetry increases the uncertainty of the electoral results and therefore reduces the power of democracy. Such settings of asymmetry fit the reality of voters. American voters were found to "base their assessments largely on the recent economic trends" (Hansen ) and unemployment rate (Dua & Smyth ), although the incumbents are not directly responsible for the periodic shocks to the economy. Gubernatorial politics turn to unpredictable disorders if the voters place too much emphasis on local conditions that governors cannot control. If the voters are all "naïve retrospective" (Alesina & Rosenthal ), then in our simulation model, the voters' judgement of the incumbent and the expectations of the challenger will not be related to the abilities and performances of the candidates: it is similar to buying bulls by judging the milk.

.
Admittedly, in politics, we are far from a complete description of the real-world gubernatorial election. Due to the absence of presidency in the model, we did not capture the e ect of federal government and presidential election on gubernatorial election. Moreover, games involving inter-provincial a airs are much more complicated than simple public good games, and the payo from inter-provincial a airs is not the only foreign factor a ecting the election: sometimes the local recognition, citizens' discontent and hostility among provinces due to the interactions between governors may change the election outcome. Even with these defects, the spatial model we establish in this paper may be helpful in understanding the evolutionary process of government performance in gubernatorial elections in various circumstances. To proceed forward, we suggest further geopolitical studies based on the framework, emphasizing the spatial features and evolution of the democratic system.