Marie Cottrell

Tatyana S Turova

Use of an Hourglass Model in Neuronal Coding

Keywords: Ergodicity, Hourglass Model, Stochastic Neural Network AMS Classi cation: 60K20, 60K35

We study a system of interacting renewal processes which is a model for neuronal activity. W e show that the system possesses an exponentially large number with respect to the number of neurons in the network of limiting con gurations of the " ring neurons". These we call patterns. Furthermore, under certain conditions of symmetry we nd an algorithm to control limiting patterns by means of the connection parameters.

Introduction

De nition of the model. We study here the hourglass model. This model is represented by a system of interacting renewal processes, numerated by the sites in a nite subset Z . De ne for any z 2 its neighbourhood: Dz : = fz l k ; k= 1 ; : : : ; K g ; 1.1 where fl k ; k= 1 ; : : : ; K g Z is a xed but arbitrary set of non-zero vectors. Notice by our de nition, that z 6 2 Dz, and for any z;y2 z 2 Dy i y 2 Dz:

Also let for any subset J DJ : = fj l k ; k= 1 ; : : : ; K ;j2 Jg:

De ne on some probability space ; ; P some random variables X i 0, Y i , ij , i 2 , j 2 Di, to be independent with densities having a certain number of moments. The variables X i 0 and Y i , i 2 , are assumed to be positive and to represent the initial state and the self-characteristic of the ith neuron, respectively. W e shall call the ij the connection parameters. For any xed i 2 , j 2 Di, the distribution of ij is assumed to be concentrated either on R , or R + . W e will specify the sign of ij in every example we treat below. Assume also ij 0 for any i 2 , j 6 2 Di. Finally let Y n i , Y n;z i and n ij , n;z ij , n 1, z 2 , be independent copies of the variables Y i and ij , respectively.

We shall de ne a Markov process Xt = X i t; i 2 , t 0, with left-continuous trajectories in R + as follows. For all i 2 and t 0 de ne X i t = X i 0 , t + X 0 tn t: X i tn=0 Y n i 1.2 + X j2Di X 0 tn t: X j tn=0

Y n;j i , X i t n IfX i t n n ji g

, n ji IfX i t n n ji g , X j2DDinfig Di X z2Di Dj: zi 0 X 0 t n t : X j t n = 0 ; X i t n n ji ; and X z t n n;j jz n;j zi : Biological interpretation. Any moment t such that X z t = 0 for some z 2 we call the moment of ring of the zth neuron. Assume for a moment that ji 0 for any i; j 2 . Then the last two summation terms in the right-hand side of equation 1.2 turn into zero, meaning that X i t is merely a renewal process in this case. Thus each component X i t represents the duration of time before the next ring of the corresponding ith neuron assuming no interaction takes place m e anwhile.

In the presence of non-zero interactions the dynamics of Xt = X i t; i 2 , t 0, are described as follows. As long as all the components of Xt are strictly positive, they decrease from the initial state X0 linearly in time with rate one until the rst time t z 1 that one of the components reaches zero for some z 1 2 : X z 1 t z 1 = 0 . We s a y that at this moment t z 1 , the z 1 th neuron res and sends impulse 1 z 1 j to the jth neuron, if j 2 Dz 1 . This means the following. At the moment t z 1 , the trajectory X z 1 t z 1 jumps to a random value Y 1 z 1 , i.e. X z 1 t z 1 + = Y 1 z 1 ; 1.3 which corresponds to the rst summation term in 1.2. At the same moment every trajectory X j t with j 2 Dz 1 receives an increment 1 z 1 j independent of the other processes, more precisely, according to the second summation term in 1.2 X j t z 1 + = 8 : X j t z 1 , 1 z 1 j ; if X j t z 1 , 1 z 1 j 0; Y 1 j ; otherwise. 1.4 Notice, that by 1.4 the negative connections 1 z 1 j delay the moment when X j t hits zero, i.e. the moment of ring of the j-th neuron. Whereas positive connections can shorten the time-interval until the next ring. That is why we shall call negative connections inhibitory and the positive ones excitatory.

Case 1. Assume that all the interactions ij , j 2 Di, are negative. Then de nition 1.2 simply becomes X inh i t = X i 0,t+ X 0 tn t: X i tn=0

Y n i + X j2Di X 0 tn t: X j tn=0 j n ji j; 1.5 for i 2 and t 0. W e use a notation X inh t for this particular case. This model is due to Cottrell 1992 1 and it has been intensively studied see an account of the previous results in Section 2 below.

Case 2. Assume, some of the interactions ij are positive. Notice that, only if z 1 j 1 0, i t m a y happen that X j 1 t z 1 , 1 z 1 j 1 0. In this case the trajectory X j 1 t is reset instantaneously to an independent v alue Y 1 j according to 1.4. Furthermore, we s a y that the j 1 th neuron also res at the moment t z 1 and changes instanteneously the trajectories of the neighbouring neurons according to the last term in 1.2, which means the following. Given X z 1 t z 1 = 0 , de ne the set of the ring at the same moment t z 1 neurons: F 1 z 1 ; t z 1 : = fj 1 2 Dz 1 : X j 1 t z 1 , 1 z 1 j 1 0g:

Then at time t z 1 + the state of the system is de ned as follows:

X j t z 1 + 1.6 = 8 : Y 1 j ; if j 2 f z 1 g F 1 ; X j t z 1 , 1 z 1 j , P i 1 ij ; if j 2 Dz 1 n F 1 ; X j t z 1 , P i 1 ij ; if j 2 DF 1 n Dz 1 F 1 f z 1 g ;
where F 1 = F 1 z 1 ; t z 1 , the summation P i runs over the set fi 2 F 1 : i 2 Dj and ij 0g, and any summation over an empty set equals zero. The rest of the trajectories X i t with i 6 2 DF 1 F 1 Dz 1 f z 1 g remain unchanged. After moment t z 1 the foregoing dynamics are repeated.

To end this description recall the well-known fact from physiology that a neuron does not react to the incoming impulses during a certain period right after it's own ring. This period is called a refractory period. Observe that in our model a neuron receives an impulse only in the moments when it does not re itself. This re ects the property of the refractory period.

This type of neural network has been proved to be equivalent, in a sense, to the "classical" neural model, which describes the interacting membrane potentials see 12 .

Plan of the paper. Our paper is organized as follows. In Section 2 we nd the critical values of the parameters which separate the ergodic and transient cases. Also we provide some historical comments on our model in Section 2. In Section 3 we analyze a fully connected network with inhibitory connections only, in which case we solve a problem related to the memory capacity.

2 Critical parameters.

Results.

Here we formulate our results on the critical values of the parameters which separate the ergodic and transient cases of the networks with excitatory and inhibitory connections.

In order to eliminate boundary e ects, we shall assume here that = f,N ; : : : ; N g is a -dimensional torus, i.e. we identify any t wo points i 1 ; : : : ; i and j 1 ; : : : ; j in Z whenever ji m , j m j 2 f 0; 2Ng for every 1 m . Let 0 be the subset of the points in such that: i the origin 0; : : : ; 0 2 0 ; ii kx , yk 6 = 1 for any x; y 2 0 :

Consider now the network Xt = X i t; i2 de ned in 1.2 with the following connection architecture.

Assumption 1 Let the neighbourhood Di de ned in 1.1 be such that Di = D I i D E i; where D I i = fj 2 : ki , jk = 1 g and D E i 0 i i 2 0 ; D E i n 0 i i 2 n 0 ; where ij k ; i6 = j are independent copies of positive independent random variables k , respectively, k = 1 ; 2; with E k = 1 , and where w I and w E are positive parameters of the inhibitory and excitatory connections, respectively.

In words the condition 2.2 means that the nearest connections are inhibitory while the more distant ones are excitatory.

Assume further that X0; X i 0; i2 , are i.i.d., and also Y; Y i ; i2 , are i.i.d. with EY = 1 : Assumption 2 Assume, that the distributions of Y; 1 , 2 , and X0 have densities g 0 u; g 1 u; g 2 u and g 3 u, r espectively, which are p ositive di erentiable functions, such that for some positive constants a and g k u ae , u for all u 0 and k = 0 ; 1; 2; 3:

2.3

We note that the assumption of the exponential decay, though seemingly rather restrictive, arises naturally from the physiology see, for example 12 and the references therein. for all 0 w E C 0 .

We postpone the proofs of these theorems to the next section. So far only the case w E = 0 has been studied analytically. Cottrell 1 proved that when w E = 0 , the network Xt = X inh t see 1.5 is ergodic, whenever w I 1=2, which is, clearly a particular case of 2.6. In 1 su cient conditions for convergence and transience were found for the nite network Xt with Z 2 and a speci c structure of the connections.

Further Piat 10 1994 extended these results to a more general connection structure. [START_REF] Karpelevich | Stochastic Evolution of Neural Networks[END_REF] provided a complete analysis of the evolution of inhibitory networks when the matrix of the expected values of the connections de nes a self-adjoint operator.

Paper 2 presents simulated results for the model, in which both inhibitory and excitatory connections were incorporated. Here for the rst time we study analytically a rigorous mathematical model for such a network. Our result explains the slope of the diagram given on Fig. 4 2 i n t h e neighbourhood of the critical value 0; 1=2. The most attractive feature of the hourglass network for neuromodelling is that in the transient case the system splits as t " 1 into two subsets. These are a subset of active i.e. in nitely often ring neurons and one of inactive neurons, which can be recognized as dark and white areas, respectively, in the simulations, see e.g. gures 3 and 6 of 2 . A rigorous de nition of the possible limiting patterns of active and inactive neurons, called traps see Section 2.2.1 below was given by Karpelevich, Malyshev and Rybko in 7 . Thus the set of all possible traps for a network should be naturally thought of as a system of patterns, which the network can hold, that is, memorize and recognize, hopefully. Recently 8 obtained a result on phase transitions in the thermodynamical limit, which establishes the large memory capacity of these networks.

Further in Section 3 besides a description of the possible patterns, we solve a n i n verse problem. This is the problem of how to determine the connections in order to get a system which stores a given system of patterns.

2.2 Proofs of Theorem 1 and Theorem 2.

Preliminary de nitions and results.

We shall use in our proofs the results of 7 . Therefore rstly we i n troduce the terminology of 7 for our model.

For any non-empty W we call X W t the restriction of process Xt on the set W. This is de ned as Xt, but with initial conditions

X W i 0 := X i 0; if i 2 W; 1; if i 6 2 W;
in which case we assume that X W i t = 1 if i 6 2 W, for all t 0, i.e. the nodes n W are "deleted".

Assume, the process X W i t; i2 W is ergodic for some non-empty W

. In this case we de ne for any i 2 W and j 2 D E i W the following limiting frequencies:

W;0 i := lim T!1 1 T f0 t n T : X W i t n = 0 g; 2.7 and W;e ij := lim T!1 1 T f0 t n T : X W j t n = 0 and X W i t n , n ji 0g: 2.8
Thus W;0 i is the limiting frequency of ring of the ith neuron due to the hitting 0 by the trajectory X W i t, while W;e ij is the limiting frequency of ring of the ith neuron due to an immediate excitatory impulse from the jth neuron. Then it is natural to call the total limiting frequency of ring of the ith neuron the following sum of the de ned above limits:

W i := W;0 i + X j2D E i W W;e ij ; i 2 W:
2.9

Next we de ne the second vector eld v W = v W j ; j2 nW by the following formula for its components: v W j = ,1 ,

X i2D I j W E ij W i , X i2D E j W E ij W;0 i ; j 2 n W: 2.10
Formally v W j is the limiting mean drift of the jth component X W j t. Indeed, according to our de nition 1.2 the ith neuron sends inhibitory impulses to the corresponding neighbours with the limiting frequency W i , but it sends excitatory impulses with the limiting frequency W;0 i . F or the details on the application of the theory of the second vector eld we refer to 7 and the references therein.

We shall call a t r ap for the process Xt any non-empty set M such that the process X nM i t; i2 n M is ergodic, while any coordinate of the second vector eld v nM is positive, i.e. v nM j 0 for all j 2 M:

2.11 For future reference let us rewrite now in our notations the criteria from 7 on inductive ergodicity and transience conditions.

Theorem A See Theorem 2.1, 7 . I. Inductive ergodicity. If for any W the process X W i t; i2 W is ergodic and v W j 0 for all j 2 n W then the process Xt is ergodic.

II.Su cient transient conditions. If there exists a set M which is a trap for the process Xt, then the process Xt is transient.

Subsystems with excitatory connections only.

Consider now the restriction X 0 t. Notice that due to the de nition of 0 and Assumption 1 the only connections between the components of X 0 i t; i2 0 are excitatory. Then by the de nition of the process X 0 t we h a ve for all i 2 0 :

X 0 i t = X i 0 , t + X 0 sn t: X 0 i sn=0 Y n 2.12 , X j2D E i X 0 tn t: X 0 j tn=0 w E n;j 2 IfX 0 i t n w E n;j 2 g +X 0 i t n , Y n;j IfX 0 i t n w E n;j 2 g ;
where n;j 2 , n 1; j 2 0 , are independent copies of the variable 2 . Clearly, for any i 2 0 the one-dimensional X fig i t is ergodic. Further for any w E 0 and for any W 0 such that X W i t; i2 W is ergodic, the second vector eld de ned in 2.10 has negative coordinates only:

v W j = ,1 , X i2D E j W w E W;0 i ; j 2 0 n W: 9 Cahiers de la MSE -1999.24
Hence, by Theorem A on inductive ergodicity X 0 i t; i2 0 is ergodic for any xed w E 0. T aking into account translation invariance, we derive from 2.12 and 2.9 0 i =: + w E = +;0 w E + K E +;e w E 0 2.13 for any i 2 0 . In particular, + 0 = +;0 0 = 1 =EY = 1 ; 2.14 since in the case w E = 0 we h a ve a system of independent renewal processes numerated by the sites of 0 .

Thus one can view the system Xt as a result of inhibitory interactions between two independent at initial moment excitatory networks: X i t; i 2 0 and X i t; i 2 n 0 .

Proof of Theorem 1.

We shall show that under condition 2.4, with

w cr I w E = 1 2 + w E 2.15
the set 0 is a trap for the system Xt. Then the rst statement on transience in our theorem follows from Theorem A.

As we h a ve shown above, X 0 t; i 2 0 is ergodic. Computing the second vector eld with respect to formula 2. face W the components of the second vector eld v W de ned by 2.10 are negative. It obviously follows from our de nition of the model, that the limiting ring frequency is the highest when the neuron has the excitatory connections only and the maximal number of them, i.e. W i 0 j = + w E for any W 2 , a n y i 2 n W and j 2 n 0 . F rom here and 2.10 we derive for any i 2 n W

v W i , 1 + 2 w I + w E 0;
where the last inequality is due to 2.5. Hence, using Theorem A Theorem 2.1 7 on inductive ergodicity, w e readily derive our statement on ergodicity. Theorem 1 is proved.

Proof of Theorem 2.

Let p A t; u A , u A 2 R + jAj , denote for any nite A 0 the density of the process X 0 A t = X 0 i t; i2 A. In the particular case w E = 0 , w e denote the corresponding density b y p 0 A t; u A . Due to ergodicity the following limit lim t!1 p A t; u A = : p A u A 2.18 exists for any A 0 . In particular, we h a ve for the limiting density o f a n independent renewal process: lim t!1 p 0 A t; u A = p 0 A u A = Y j2A p 0 j u j 2.19 where p 0 j u = R 1 u g 0 vdv EY =: p 0 u:

2.20 Further we will use the following lemma stating that the limiting density in 2.18 is a small perturbation of the function in 2.19 when the parameter of interaction w E is su ciently small. Lemma 1 There exist positive and independent of N constants C; and c such that for any 0 w E c and for any subset A 0 jp A t; u A , p 0 A t; u A j w E C jAj e , P i2A u i ; 2.21 for any t 0 and u A 2 R A + .

11

Cahiers de la MSE -1999.24

Proof. Recall that g 0 and g 3 are the density functions of the variables Y and X0, respectively. Let pt; u 0 ; v 0 , u 0 ; v 0 2 R + 0 , denote the transition density of the process X 0 i t; i 2 0 . Then for any A 0 the density p A t; u A , u A 2 R + jAj , i s g i v en by the formula p A t; u A = Z

R+ j 0 nAj Z R+ 0 0 @ Y z2 0 g 3 u z 1 A pt; u 0 ; v 0 d u 0 d v 0 nA : 2.22
In the particular case w E = 0 , w e denote the transition density b y p 0 t; u 0 ; v 0 . Clearly, in this case p 0 t; u 0 ; v 0 = Y j2 0 p 0 t; u j ; v j due to the independence of the components. Notice, that the transition density of each of the independent renewal processes is p 0 t; u; v = 2.23 = u , t , v; if 0 t u , g 0 t , u + v + P 1 k=1 R t,u 0 p S k xg 0 t , u + v , x dx; if 0 u t, for every u; v 2 R + and t 0, where u , is the Dirac distribution concentrated at u; Y;Y k ; k 1, are i.i.d.; S k := P k l=1 Y l ; k 1, and p S k is the density of the distribution S k .

Using the results of Stone 11 one can show that under Assumption 2 there exist positive constants C 0 and 0 such that jp 0 t + t 0 ; 0; v , p 0 t; 0; v j C 0 e , 0 t+v 2.24 for all v 2 R + and t; t 0 0 for the details see for example, 13 Section 4.

Combining 2.24 and 2.19, we get: jp 0 v , p 0 t; 0; v j C 0 e , 0 t+v :

2.25

For the general theory of piece-wise linear stochastic processes we refer to the book 3 b y D a vis, 1993.

After these preliminaries the proof of Lemma 1 follows by standard techniques using so-called cluster expansions see, e.g. 13 for a similar model and 6 for a class of stochastic processes with local weak interactions as soon as the Kolmogorov equation see 2.26 below is written down for the transition density pt; u 0 ; v 0 .

Let M R 0 + denote a class of densities on R 0 + which possess all continuous partial derivatives. De ne also pt; u 0 ; , = Z , pt; u 0 ; v 0 dv 0 for any Borel set , 2 R 0 + . Denote here Oi = D E i and w E = . Then for our model we derive for any density 2 M R 0 + and for any Borel set

, 2 R 0 + @ @t Z R 0 + u 0 pt; u 0 ; , du 0 2.26 = , Z R 0 + X i2 0 u 0 @ @u i pt; u 0 ; , du 0 + R R 0 + u 0 P i2 0 u i R R 1+2jOij + p Y w i Q j2Oi p Y y j p 2 w j
pt; u 0 + U 0 u 0 ; w i S Oi ; y Oi ; , , pt; u 0 ; , d y Oi d w i S Oi du 0 ;

where is the Dirac measure concentrated at 0, p Y u = g 0 u; p 2 u = g 2 u= = ; if 0; u; if = 0 ; u 0; and the vector U 0 u 0 ; w i S Oi ; y Oi has components: U j u 0 ; w i S Oi ; y Oi = 8 :

w i ; if j = i;
,w j Ifw j u j g + y j , u j Ifw j u j g; if j 2 Oi; 0; otherwise.

Further for any w i 0 let the vector w i = w i j ; j2 0 have the following components: w i j = w i ; if j = i; 0; otherwise.

=: Z R 0 + u 0 X i2 0 H i; 0 0 + H i; 0 1 pt; u 0 ; u 0 :
Notice, that when = 0 the operator H i; 0 1 0 for any i. In this case 2.27 simply becomes @ @t Z R 0 + u 0 p 0 t; u 0 ; , du 0 = Z R 0 + u 0 X i2 0 H i; 0 1 p 0 t; u 0 ; , d u 0 2.28 i.e., the equation for the density of the process whose components are independed renewal processes. Hence the formula 2.23 gives a solution to 2.28: p 0 t; u 0 ; , = Z , p 0 t; u 0 ; v 0 d v 0 : Now w e can nd the solution to equation 2.27 by solving the equivalent integral equation for the reference see also 13

;i k 2 k 0 Z t 0 Z R 0 + Z R 0 + Y z2 0 g 3 u z p 0 t , s 1 ; u 0 ; u 1 0 H 1 ; 0 1 Z s 1 0 Z R 0 +
p 0 s 1 , s 2 ; u 1 0 ; u 2 0 : : :

H i k ; 0 1 Z s k,1 0 Z R +
p 0 s k ; u k 0 ; v 0 dv 0 nA du k 0 ds k : : : d u 1 0 du 0 ds 1 ;

Making use of the formula 2.23 and the bounds 2.3 and 2.24, one can prove the convergence of the series in 2.30 and get the necessary bounds by following the the formulae in Section 3 of 13 p.179 and further with only minor modi cations due to 2.26. Therefore for the sake of brevity w e skip the rest of the straightforward part of the proof of Lemma 1.

Consider now the embedded Markov c hain x i n : = X 0 i n ; i2 0 , n = 1 ; 2; : : : ;where 1 ; 2 ; : : :denote the consecutive moments when at least one of the components of X 0 reaches zero, i.e. X 0 j n = 0 for some j 2 0 .

Due to ergodicity o f X 0 i t; i 2 0 , there exists a limiting distribution, call it F, o f xn as n ! 1 . Let x 1 n be the stationary version of the Markov chain xn, i.e. whose initial distribution is F. F urther for any i 2 0 and j 2 D E i let Z ij be a random variable with distribution function F ij u : = Pfx 1 i n u j x 1 j n = 0 g:

Clearly, the density function f ij for each F ij is given by f ij u = p fijg u; 0 p j 0 2.31 as long as p j 0 0. Observe that p 0 j 0 = 1 EY = 1 by 2.20. Hence, according to Lemma 1 condition p j 0 0 is satis ed at least for all small values of w E .

We shall derive n o w an equation for 0 ;0 i = +;0 w E see de nition 2.7 and 2.13 . From 2.12 and the above arguments on ergodicity w e obtain for all i 2 0 :

lim t!1 1 t X 0 i t = 0 = ,1 + +;0 w E EY , K E +;0 w E w E 2.32 , X j2D E i +;0 w E EZ ij , Y , w E 2 IfZ ij w E 2 g;
where Z ij , Y and 2 are independent b y their de nitions. Therefore taking into account that EY = 1 , w e obtain from 2.32 the following equation:

+;0 w E 0 @ 1 , K E w E + X j2D E i PfZ ij w E 2 g + X j2D E i Ew E 2 , Z ij IfZ ij w E 2 g 1 A = 1 : 2.33
Lemma 2 There exists a positive constant C 1 independent of N such that for any i 2 0 and for any j 2

j + w E , 1 + K E w E j w 2 E C 6 2.47
for all w E C 0 , where C 6 and C 0 are some positive constants.

Substituting the bound 2.47 into 2.15 we get 2.6. This completes the proof of Theorem 2.

3 Fully connected nite networks.

3.1 The model. Let = f1; : : : ; N g. Consider an N-neuron network with inhibitory connections only, i.e. Xt = X inh t as de ned in 1.5. Here we assume that Di = nfig for any i 2 , which means that our network is fully connected.

Let N = 2 pk, where k;p 2 Z + are xed arbitrary. Suppose a network consists of 2p symmetric subnets blocks, determined by the values of the connection constants E ij as follows. We divide set = f1; :::; Ng into 2p subsets W i , i = 1 ; :::; 2p, so that jW i j = k and 2p i=1 W i = :

3.1

Notice that we k eep k 1 xed but arbitrary. This is simply a size of one block or unit of our network. The case k = 1 is trivial.

Let us divide also the set f1; :::; 2pg into a set of p non-intersecting pairs, i.e., f1; :::

; 2pg = p n=1 C n = p n=1 fC 1 n ; C 2 n g; 3.
2 where jC n j = 2 ; and C i C j = ;: Then we set EY x = a 0; E xy = ,c ij 0; if x 2 W i ; y 2 W j ; Thus our network consists of 2p connected blocks of interacting identical neurons. The connections between the neurons of di erent blocks are different in general from those between the neurons of the same block. In the following Theorem 3 below, we describe all the possible traps for our network, i.e. all the possible limiting states in the transient case. Notice that the total number of the patterns is jTj = 2 p = 2 N 2k , which is exponentially large with respect to N. This description will allow us to nd a method to reconstruct or "memorize", in some particular cases, a given system of traps see Corollary 1 below. More precisely, w e will show h o w t o c hoose the appropriate connection constants. L i C n = 1 ; i = 1 ; 2; n = 1 ; :::; pg: Then a set A is a trap for the network with the connection constants 3.3-3.5, if and only if A = fx 2 : x 2 i2B W i g; Proof. First we will show that any set A satisfying conditions 3.7 and 3.8 is a trap for the above de ned system. Indeed, consider a restriction X nA t. In this case X nA i t; i 2 n A is a completely connected system, and for any x; y 2 n A, y 6 = x ,E xy = b a = EY x ; 3.17 where fl;l 0 g = f1; 2g and sets I and I 0 are such that I I 0 6 = ;; I I 0 = ;; and I I 0 J = f1; : : : ; 2pg; 3.18 II W i n A i 6 = ; at least for some i 2 B, i.e. 3.7 is not satis ed.

Suppose situation I takes place. Without loss of generality let l = 1 and l 0 = 2 . Consider

A := n A = f i2 B W i g f i2B W i n A i g: 3.19
De ne the subset B 1 B so that W i n A i 6 = ; i i 2 B 1 ;

3.20 and thus

A = f i2 B W i g f i2B 1 W i n A i g:
Notice that B 1 can be empty. F urther de ne I 1 so that C n 2 B B 1 i n 2 I 1 :

3.21 a Suppose I 1 6 = ;. Let B 0 := fC 1 n ; n 2 I 1 g f B B 1 n f C n ; n 2 I 1 gg:

3.22 Denote A B 0 = f i2 B B 0 W i g f i2B 1 B 0 W i n A i g:
Then the subsystem X A B 0 i t; i 2 A B 0 is ergodic, since it is completely connected, and 3.9 holds for any x; y 2 A B 0 , x 6 = y. Also, we can compute as in 3.13

A B 0 x = 1 a + j A B 0 j , 1b ; 3.23
Taking into account that jI 1 j 1, w e easily derive from 3.23 the following upper bound for the components of the second vector eld v A B 0 : v A B 0 z , 1 + c + j A B 0 j , 1b a + j A B 0 j , 1b 0

22

Cahiers de la MSE -1999.24

for any z 2

An A B 0 due to condition 3.5. Hence, X A i t; i 2

A is transient according to Theorem A. This contradicts our assumption that A is a trap.

b Suppose now that I 1 = ;. Then it follows from 3.21 and 3.17 that we also have I 0 = ;:

3.25

This together with our assumption 3.18 implies that I 6 = ;. T h us in this case we h a ve A = fx 2 : x 2 i2B A i g;

3.29

The latter contradicts our assumption, that A is a trap. Hence we conclude that if A is a trap then necessarily condition 3.8 holds. Assume now situation II. More precisely, suppose that A satis es 3.8 but does not satisfy 3.7, i.e. A = fx 2 : x 2 i2B A i g; 3.30 for some B such that 3.8 holds, while W i n A i 6 = ; for i 2 B 1 B for some nonempty B 1 see de nition 3.20 . The latter implies

A = f i2 B W i g f i2B 1 W i n A i g;
where according to the assumption 3.8 jB 1 C n j = 1

23

Cahiers de la MSE -1999.24

for at least one C n . The latter implies C n 2 B B 1 , i.e. the set I 1 de ned in 3.21 is non-empty. But as we h a ve seen, this situation contradicts the assumption that A is a trap. This nishes our argument that conditions 3.7 and 3.8 are necessary for a set A to be a trap. This completes the proof of Theorem 3.

Learning rule for the almost symmetric case

We will show here that the Hebbian rule see 5 of learning patterns which successfully works in the case of Hop eld networks 5 , is applicable for our network at least for the following particular case.

For any A de ne con guration A = i A; i 2 2 f , 1; +1g such that i A = +1; if i 2 A; ,1; otherwise.

3.32

We shall also call a con guration a trap for Xt if and only if = A, where A is a trap. Suppose that we are given 2 p , p = N 2k , binary vectors images 1 ; : : : ; 2 p. We shall nd the connection constants E xy such that Xt satisfying 3.3 with these parameters, possesses a system of traps consisting exactly of the given 2 p vectors.

Notice the di erence between this task and the problem of stability o f patterns for Hop eld neural model see for example, 9 for a recent account on the relevant results. Recall, that the capacity of Hop eld network is determined by the number of given i.i.d. patterns, which are stable with respect to the dynamics of the system. This means that starting from an arbitrary initial state the system should converge with a large probability to one of the given patterns, which is the closest to the initial state. It was conjectured that the number of such patterns for Hop eld network of N neurons is at most a fraction of N. Here we construct a network which posesses given exponentially large with respect to N n umber of the limiting patterns and only them. These patterns are stable in a trivial sense, i.e. if the initial state of the system is one of the given traps patterns, then the system stays at this trap forever. However, we do not predict which state out of p possible the system converges to, starting from an arbitrary initial condition. The problem of determination of the basins of attraction of the limiting patterns for the hourglass model will be a subject of a separate study.

24

Cahiers de la MSE -1999.24

Corollary 1 Suppose the collection of N-dim vectors f ; = 1 ; : : : ; M g where M = 2 p , and x 2 f , 1; 1g for any and 1 x N, has the following properties:

1.

x y = 1 for any if x; y 2 W n for some n 2 f 1; :::; 2pg, 2. P N x=1 x = 0 for all , 3. for any n 2 f 1; :::; 2pg there exists unique l = ln 2 f 1; :::; 2pg n f ng such that x y = ,1 for any whenever x 2 W n and y 2 W l .

Then the N-neuron system with a i = a and the connection constants E xy = bx; y; if bx; y = min x 0 ;y 0 bx 0 ; y 0 max x 0 ;y 0 bx 0 ; y 0 ; otherwise, [START_REF] Davis | Markov Models and Optimization Chapman and Hall[END_REF]

3 . 3 for

 33 all x; y 2 and 1 i; j 2p, where c ij = c; if fi; jg = C n for some n;

3. 2

 2 Description of patterns. Theorem 3 Let N = 2 pk, where k 1; p 1, and decompositions 3.1 and 3.2 be xed but arbitrary. Call L the set of all the decompositions of f1; :::; 2pg into a pair of subsets such that L := fL 1 ; L 2 : L 1 L 2 = f1; :::; 2pg; L 1 L 2 = ;; 3.6

  conditions 3.4-3.5 are satis ed. Hence we can use Theorem 3 to conclude that f , = 1 ; : : : ; 2 p g is a system of traps for the de ned network.12 T.S. Turova, Analysis of a biologically plausible neural network via an hourglass model. Markov Processes and Related Fields 2 1996 487-510. 13 T.S. Turova, Exponential rate of convergence of an in nite neuron model with local connections. Stochastic Processes and Their Appl. 73 1998 173-193. 27 Cahiers de la MSE -1999.24

  D E i jPfZ ij w E 2 g , w E j w 2

	E C 1 Ew E 2 , Z ij IfZ ij w E 2 g w 2 E C 1 for all 0 w E c with c de ned i n L emma 1. and Proof of Lemma 2. In the case w E = 0 let Z ij = Z 0 ij . Then the density o f 2.34 2.35 Z 0 ij call it correspondingly f 0 ij u i s g i v en by f 0 ij u = p 0 u = u g 0 ydy 2.36 First let us obtain the following bound as a corollary of Lemma 1: jf ij u , f 0 ij uj w E C 2 ; u 2 R + ; 2.37 = Z 1 0 g 2 y 0 w E y , uf 0 ij u du dy Z w E y Ew E 2 , Z ij IfZ ij w E 2 g due to 2.31 and 2.20. To prove 2.35 let us consider the following decomposition: 2.39 gives us 2.34. Combining the bounds 2.42 and 2.41 together with equations 2.40 and Z 1 where C 2 is some positive constant independent o f N. Notice, that from + Z 1 0 g 2 y Z w E y 0 w E y , uf ij u , f 0 ij u du dy: 2.21 and 2.18 we immediately derive: jp A u A , p 0 A u A j w E C 3 expf, X i2A u i g; Using again 2.3 and 2.37 we immediately derive from here: 2.38 Ew E 2 , Z ij IfZ ij w E 2 g w 2 E C 3 + w 3 2.43 E C 4 ; for any u A 2 R A + , where C 3 is some positive constant independent o f N. Then the bound 2.37 readily follows from 2.38 and the continuity of the transformation 2.31. where C 3 and C 4 are some positive constants, which implies 2.35. The lemma is proved. Lemma 2 allows us to derive from equation 2.33 that Let us prove n o w 2.34. Consider PfZ ij w E 2 g = Z 1 0 g 2 y Z w E y 0 f 0 ij udu dy j +;0 w E , 1j w 2 E C 5 2.44 2.39 + Z 1 0 g 2 y Z w E y 0 f ij u , f 0 ij u du dy: for some positive constant C 5 . Finally let us consider +;e ij w E for i 2 0 ; j2 D E i: According to de nition 2.8 we derive similarly 2.32 Substituting formula 2.36 into the rst integral in 2.39 we derive after simple calculations: Z 1 0 g 2 y Z w E y 0 f 0 ij udu dy W;e ij = lim t!1 1 t f0 t n t : X W j t n = 0 and X W i t n , n ji 0g 2.45 2.40 = Z 1 0 Pf 2 ygw E 1 , PfY w E yg dy = w E , w E = lim t!1 1 t X IfX 0 i t n w E n 2 g 0 tn t: X 0 j tn=0 Z 1 = +;0 w E PfZ ij w E 2 g: 0 Pf 2 ygPfY w E yg dy; where we used the condition E 2 = 1 . T aking into account 2.3 it is easy to see that w E Z 1 0 Pf 2 ygPfY w E yg dy C 0 w 2 E 2.41 Substituting 2.45 into de nition 2.13 we obtain + w E = +;0 w E 1 + X PfZ ij w E 2 g; 2.46 j2D E i for some positive constant C 0 . Next, making use of 2.37 we readily derive j Z 1 0 g 2 y 0 f ij u , f 0 ij u du dyj w 2 E C 2 : 2.42 Z w E y which together with 2.44 and 2.34 gives us the following bound:

  of traps, which is f , = 1 ; : : : ; 2 p g. Proof. Indeed, having conditions 1-3 of the corollary satis ed, we derive Ba b x; y ,B , Aa otherwise. 3.37 Substituting 3.36 snd 3.37 into 3.33, and taking into account condition 3.35, we obtain ,E xy = A + Ba a ;if x 2 W n ; y2 W ln ; ,E xy = B , Aa a ;otherwise,

				.33
	where with the constants A and B satisfying the conditions bx; y : = Aa 1 M M X x y , Ba; =1	3.34
			0 B , A 1; 1 B + A;	3.35
	has a system from 3.34 that
		bx; y =	,A + Ba; if x 2 W n ; y2 W ln ; ,B , Aa; if x; y 2 W n ;	3.36
	and	,A +

Cahiers de la MSE -1999.24

Cahiers de laMSE -1999.24

Acknowledgements We thank the referee for useful remarks.

25

Cahiers de la MSE -1999.24


Jean-Philippe Venot
email: jean-philippe.venot@ird.fr

Casper Bruun Jensen

Etienne Delay

William's Daré

Mosaic glimpses: Serious games, generous constraints, and sustainable futures in Kandal, Cambodia

Keywords:

This paper uses a series of serious games -a form of participatory modelling designed and played in Kandal, Cambodia -as an entry point for reexamining relations between development projects, participatory formats, landscape transformations, and sustainable futures. Critics of development and participation have shown that participatory formats simplify real-world complexities by rendering them technical. This is also the case for serious games. But contrary to what is often assumed, 'depoliticization' is not the unavoidable outcome. Instead, participatory outcomes depend on specific sociotechnical patterns of more or less generous constraints. To support collective exploration requires tinkering with these patterns of constraints to keep the boundaries between virtual and real worlds, insiders and outsiders, and the present and future relatively permeable. Generous constraints and permeable boundaries do not keep power out of participation but facilitate glimpses of different possibilities. In Kandal, they made it possible to shift from narrow technical discussions on the rehabilitation of specific preks (water channels) towards a collective exploration of sustainable futures for the full mosaic landscape. In general, we argue, serious games hold potentials as experimental systems, which are serious to the extent that they work like technologies of humility. In this capacity, they can support efforts to do difference together, and explore more-than-human worlds and divergent practical ontologies. Learning from this multiplicity matters for moving towards sustainable forms of living in Kandal and elsewhere.

I would just like to know where you are at as far as the preks are concerned . . . On this basis, I can ''pilot test" the tools we developed in the area you identified in the inception report . . . this could constitute the basis for another way to approach the preks.

Dan Bruins [senior international expert leading team 1]:

The trouble I see here is that this is not a game; there is already an experienced team that is employed to do this [identify interventions for prek rehabilitation] and is doing it. . . and it is work and a responsibility.

Jean-Philippe Venot:

I am not proposing ''we just play around." I just think that the tools we developed (even if they may appear to be not really serious) can actually help WAT4CAM implementation. . . If we do not do it now, we all know how it will end up. The pressure to ''deliver" will mean that people will do what they best know . . . Given the complexity of the preks, this will lead to very similar mistakes. Experimenting now may yield practical things that the Technical assistance team 1 and 2 will be able to act upon later.

Jan Shiller [senior international expert leading team 2]: 1) I also think the word 'game' is very misleading . . . 2) The reason I support this activity is not because I like the 'game' but purely that in my mind a master plan should definitely be fully participative in set-up, and cover all stakeholders and involved institutions . . .

Development, participation, and the problem of constraints

Since the 1990s, development and participation have both been subject to heated debate and critique. Development projects were described as dependent on actively 'unknowing' contexts and rendering all problems 'technical' [START_REF] Ferguson | Development", depoliticization, and bureaucratic power in Lesotho[END_REF][START_REF] Li | The will to improve: Governmentality, development and the practice of politics[END_REF]. This depoliticized many of the central issues and created lacunae of accountability.

While diverse participatory approaches aimed to address these problems, they gave rise to another set of critiques. Centrally, they were criticized for inattentiveness to, or naivety with respect to, inequality, oppression, and power [START_REF] Cooke | Participation: The new tyranny[END_REF]Hickey & Mohan, 2004: 11). It was also suggested that the radical emancipatory potentials of participation evaporated as these approaches became one among many instruments within the 'toolbox' of development agencies. Participation appeared as depoliticization in a more appealing guise.

Since WALRUS1 (Water and Land Resource Utilization Simulation), which was developed to facilitate interactions about water and land resource usage in the early 1970s [START_REF] Wesselow | Role-playing games in natural resource management and research: Lessons learned from theory and practice[END_REF], there have been experiments with participatory games in various ''modeling with stakeholders" approaches, including in development contexts [START_REF] Voinov | Modeling with stakeholders[END_REF]. The general aim is to facilitate collective exploration of complex problems [START_REF] Barreteau | Serious Games[END_REF] in virtual settings carefully designed to avoid, or at least minimize, existing power relations and conflicts. One of these approaches, Companion Modelling-ComMod- [START_REF] Barreteau | Our companion modelling approach[END_REF][START_REF] Etienne | Companion Modelling: A participatory approach to support sustainable development[END_REF], informed the serious games that we designed and played in Kandal.

While serious games have been developed somewhat to the side of the critical debates on participation, participatory modelers are attentive to many similar issues and their discussions contain indirect responses to several central objections [START_REF] Barnaud | Equity, power games, and legitimacy: Dilemmas of participatory natural resource management[END_REF][START_REF] Jordan | Twelve Questions for the Participatory Modeling Community[END_REF]. Some reflect on how games can be designed to avoid reinforcing conflicts and existing relations of domination [START_REF] Barnaud | Equity, power games, and legitimacy: Dilemmas of participatory natural resource management[END_REF][START_REF] Bécu | Participatory computer simulation to support collective decision-making: Potential and limits of stakeholder involvement[END_REF][START_REF] Mathevet | Contexts and dependencies in the commod processes[END_REF]. Others argue for attentiveness to the dominant roles and postures of facilitators and modelers [START_REF] Barreteau | Our companion modelling approach[END_REF][START_REF] Jonsson | How participatory can participatory modeling be? Degrees of influence of stakeholder and expert perspectives in six dimensions of participatory modeling[END_REF], or invoke ''ethics" as protection against the risk of manipulation [START_REF] Barnaud | La modélisation d'accompagnement pour une gestion concertée des ressources renouvelables en Thaïlande[END_REF][START_REF] Halbe | A methodological framework to support the initiation, design and institutionalization of participatory modeling processes in water resources management[END_REF]. A recent overview recognizes ''potential biases" and recommends identification of ''power relationships, social values and epistemologies of participants" [START_REF] Barreteau | Serious Games[END_REF]. Some even ponder whether participatory modeling will ever be 'truly' participatory [START_REF] Jonsson | How participatory can participatory modeling be? Degrees of influence of stakeholder and expert perspectives in six dimensions of participatory modeling[END_REF].

At the same time, however, two tenets are advanced about the distinctiveness of participatory modeling and serious games vis-àvis other participatory formats. The first claim about difference hinges on a displacement of where participation takes place. Because contrary to other formats, participation is not only a matter of participating in activities designed by 'outsiders' but also of involvement in the design and redesign of the participatory tools themselves. The second claim is that power relations can be prevented from corrupting serious games by carefully maintaining distance from the real world (Daré & Barreteau, 2003;[START_REF] Patamadit | The Thai traditional learning process in folk culture : Implications for the companion modelling approach[END_REF]. This is what allows the games to 'work' as participatory experiments.

It follows from these two 'central dogmas' that critiques that might well apply to other kinds of participation are irrelevant for serious games. The problems of simplification and domination involved in ''rendering technical" have been solved by involving participation at the level of design and by loosening power relations through 'distance.' (WASP 2014(WASP -2019)). The email exchange happened 2 years after a prefeasibility study had been submitted to AFD, since it took 18 months to finalize the project and recruit the technical assistance teams. None of the authors have had any economic relation with AFD or either of these projects.

3 Between 2016 and 2019, the authors received funding from the French National

Research Agency (ANR) and the Japan Society for the Promotion of Science (JPSP) (see https://delta.hus.osaka-u.ac.jp/ and https://deltasoutheastasia-doubt.com/). Research activities involved collaboration with Cambodian partners from the Royal University of Agriculture (RUA) and the NGO Irrigation Service Center (ISC). They also came to include frequent informal interactions with AFD's staff and technical teams. 4 We use the terms 'games', 'serious games' and 'participatory games' more or less interchangeably. The games are said to be serious because they deal with 'real world issues' and to be participatory because multiple actors are involved in design and play. We do not use the common term 'role-playing game,' because it focuses attention on a single element of game dynamics -the roles that players assume. 5 Hybrid forums are open spaces where heterogeneous groups (from experts to laypeople) can come together to discussion technical options and decisions that collectively affect them (Callon et al., 2001: 18). Sheila [START_REF] Jasanoff | Technologies of humility: Citizen participation in governing science[END_REF] distinguishes between modes of expertise as either technologies of hubris or humility depending on whether they focus on the known at the expense of the unknown and avoid challenges to their framing assumptions (239) or grapple with ''the unknown, the uncertain, the ambiguous, and the uncontrollable" (227).

Participatory critics might well reply that these precautions are not sufficient to immunize serious games against charges of depoliticization. After all, there are still implicit, framing assumptions, the games still simplify complex problems, and they render those problems technical in many ways. For example, although participants are given room to question and even modify game design (Daré & Barreteau 2003;[START_REF] Patamadit | The Thai traditional learning process in folk culture : Implications for the companion modelling approach[END_REF], it happens against a backdrop of ideas, rules, and more or less explicit objectives defined by researchers or development experts during preparatory workshops where the participants are often absent.

The present inquiry is distinguished both from the critiques of development and participation that center on depoliticization and from the participatory gaming literature that emphasizes the potentials of the approach. For the former, it often seems unnecessary to examine participatory processes and effects in much detail because it is (by now) taken as given that ''rendering technical" invariably results in depoliticization. The latter is seldom interested in the politics of technicalities and assumes that formal companion methodologies guarantee participatory ends. These are very different perspectives but, in consequence of their shared lack of curiosity about the socio-technical patterns of constraints that actually shape participatory processes and outcomes, they end up as strange bed-fellows. As interest in those patterns and their effects are evacuated from both sides, serious games remain tightly black-boxed.

Rather than deflecting the participatory critiques, this paper embraces them. But it does so only as a starting point, in order to move elsewhere. Thus, we affirm that serious games undeniably formalize, simplify, and 'render technical' in many different ways, including by constraining players. But this does not unavoidably compromise their participatory potentials. It just means that those potentials, rather than general and methodological, hinge on the specific patterns of constraints at play in the games. Ironically, it follows that critiques based on 'what we all know' about simplification and depoliticization, elite capture and marginalization, etc. in general are too reductive and formulaic to have much purchase.

In the rest of this article, we take the serious games designed and played in Kandal from 2016 onwards-around the same time as AFD engaged in prek rehabilitation-as an entry point for tackling relations between development agendas, "rendering technical," and participatory politics in a way that avoids the oppositions sketched above. We explore the games as hybrid forums, which can potentially support collective deliberations and speculations about landscapes and livelihoods. To understand those potentials and their obstacles, we enter the black-box and inspect patterns of more-orless generous constraints [START_REF] Gomart | Towards generous constraint: Freedom and coercion in a French addiction treatment[END_REF][START_REF] Gomart | Surprised by methadone: In praise of drug substitution treatment in a French clinic[END_REF].

We begin by describing negotiations with different partners about several kinds of constraints relating to the design of the game board. We briefly depict how social constraints and power relations entered game sessions. But we are particularly interested in other glimpses of possibility also elicited in the games. Those glimpses keep alive the possibility that serious games can work as technologies of humility even in difficult development contexts.

Generous constraints and permeable boundaries

Over the last decade, some researchers sought to loosen the stark contrasts and dichotomies that structured the early controversies around participation (e.g. Yarrow & Venkatesan, 2012;[START_REF] Jensen | Monitoring movements in development aid: Recursive partnerships and infrastructures[END_REF]. The anthropologist Maia Green (2010: 9) pointed out that the critiques were often less about participation as an ideal than about failure to uphold it in the context of specific projects and power relationships. Despite widespread recognition that participatory methods do not live up to the ideal in practice, she also noted their continued popularity, both among development organizations and aid recipients.

Green ascribed this popularity to the capacity of participatory forms to operate as ''boundary objects" [START_REF] Star | Institutional ecology. ''Translations'' and Boundary Objects: Amateurs and professionals in Berkeley's museum of vertebrate zoology[END_REF]; more or less tangible devices that facilitate temporary collaborations between different communities. In fact, participatory modelers also occasionally characterize their artefacts and techniques as boundary objects (Voinov et al., 2016: 13). There are some important differences, however.

Participatory modelers work with a normative, methodological framework in which games must be boundary objects because they are designed with diverse participants and divergent viewpoints can be in principle kept intact due to 'distance' from reality and its power relations. In contrast, [START_REF] Star | Institutional ecology. ''Translations'' and Boundary Objects: Amateurs and professionals in Berkeley's museum of vertebrate zoology[END_REF] empirically examined what allowed particular things to actually operate as boundary objects within particular collaborative contexts.

This non-normative approach paves the way for exploring serious games not only as part of heterogeneous networks (of e.g. funding agencies, project documents, technical experts, researchers, NGOs, government partners, farmers and fishermen) [START_REF] Barreteau | Companion modelling: A method of adaptive and participatory research[END_REF] but as networks in their own right. Serious games appear as open-ended experimental systems populated by players who are constrained in numerous-more or less technical and non-technical-ways. Some constraints relate to technical features and 'pre-defined scenarios.' Some concern procedures and rules. Others relate to the practical and social conditions under which games are designed and played and to the 'objectives' assigned by facilitators based on their 'expert knowledge.' And these diverse constraints by design are complemented by yet others, which are negotiated by players or emerge haphazardly during game sessions.

To critics, this will sound as another version of simplifying complex, power-laden social realities by rendering them technical. But paradoxically this is a simplified and reductive understanding of constraints. For as we shall see, constraints are not necessarily repressive, and not in the same ways. They are not always introduced by the most powerful agents. And while they do simplify, they are not only reductive.

In support, we might recall Paul Feyerabend's (1993: 72) observation that scientific progress itself depends on constraints put in place by a ''specified and well-entrenched" framework. But we draw specific inspiration from the concept of generous constraints developed by the medical sociologist Emilie [START_REF] Gomart | Towards generous constraint: Freedom and coercion in a French addiction treatment[END_REF][START_REF] Gomart | Surprised by methadone: In praise of drug substitution treatment in a French clinic[END_REF] in elaboration of Michel Foucault's famous analyses. [START_REF] Foucault | The birth of the clinic: An archaeology of medical perception[END_REF][START_REF] Foucault | Discipline and punish: The birth of the prison[END_REF], of course, was fascinated by how institutional settings shape configurations of knowledge and power. Sick bodies are organized in hospitals, students are made obedient at school, and soldiers are disciplined in barracks. But as Gomart (2004: 99) reminds us, even while soldiers become ''docile," they also become more ''competent." School children are rendered obedient, yet at the same time they become more ''vigorous" (99).

This makes little sense if one thinks of 'freedom' as the absence of external obstacles to action. If agency is diminished whenever an exterior force impinges on the subject, institutional ensembles like schools and prisons-or participatory development formatscan only appear as spaces of coercion, oppression, or subjugation. But if agency is viewed as emergent, distributed and shareable, the situation looks different. Once it is recognized that agents are always constrained, the obsession with whether or not that is the case fades in importance. Instead, what matters is careful examination of the relational qualities of specific patterns of constraints: do they merely isolate and reduce, or might they also engender new associations and possibilities for change? In place of a general critique of technical simplifications we are encouraged to tinker, add, and vary constraints in an experimental search of conditions that might generously ''do a little more than oppress the user" (Gomart 2002: 517).

Across very different empirical settings, there is broad alignment between these characterizations and Maia Green's observation that 'local people' often use participatory forms productively to achieve their own ends, including ends of self-transformation. As we continue to discuss, refraining from reducing constraints to their repressive aspects makes it possible to understand the reciprocal construction of games and participants, objects and subjects.

In our case, this also requires grappling with the permeability of boundaries. 6 Because, contrary to the introductory email exchange, which proceeded on the assumption that a single, rigid line separates unserious games from the real and serious development project, we are confronted with several power-laden and transformable relations.

Some anthropological analyses provide a vantage point for engaging these issues. Clifford [START_REF] Geertz | Deep play: Notes on the Balinese cockfight[END_REF] famously analyzed Balinese cockfights as involving spectators and gamblers in ''deep play." These fights do take place, cocks maim and kill each other, and substantial amounts of money are gambled away. This really happens, but within a highly circumscribed game space where regular conventions of behavior are suspended and actions have no significant external repercussions. Yet, despite the lack of 'real con-sequences,' the games are very serious in moral and symbolic terms. For contrast, consider, Gregory Bateson's (1987: 59) discussion of dogs playfully fighting. The constraints of the play world prevent the dogs from seriously harming each other, but the fighting skills they acquire are real and serious, and they stay with them long after play has ended. Both cases exhibit the creative potentials of temporary immersion in a 'virtual world.' They make clear that a boundary is prerequisite for there to be 'a game' but also that their 'seriousness' depends on it being neither too fixed nor too rigid.

A similar premise undergirds serious games, which aim to create ''synergy between real and virtual contexts, the one highlighting the other and vice-versa" [START_REF] Daré | A role-playing game in irrigated system negotiation: Between play and reality[END_REF]. But this cuts several ways. For companion modelers, games are serious because they make it possible to address complex real-world problems and trigger collective learning. Speculative leeway is facilitated by ensuring that there is enough distance between game scenarios and reality to prevent the latter from overdetermining the former. In contrast, as we shall see, participants often view games as serious if they resemble reality sufficiently to make it possible to voice real concerns and oppose developments that seem only too likely to happen.

This means that the question of generous constraints is somewhat more complicated than in Gomart's original case of drug users entering a clinic. This scene was characterized by two general kinds of subject positions-staff or user-and a loosely shared trajectory towards rehabilitation. Although negotiated and varied in practice, the constraints were almost exclusively introduced by the clinic. In contrast, the serious games played in Kandal involve permeable boundaries, numerous incongruent subject positions, and widely divergent interests in prek rehabilitation. Moreover, while most constraints were initially defined by researchers and their partners, they were added to and modified by players, who showed considerable creativity in making the constraints more generous for themselves.

For serious games to have a chance to operate as technologies of humility for envisioning different kinds of more-than-human landscapes, those tensions and complications have to be continuously handled. Collective speculations and differently oriented explorations must be allowed to mingle, and leakage into the landscape must be supported in full recognition that everybody is not in(to) the process for the same reasons.

Discovering the preks

At the Cambodian capital Phnom Penh, the Bassac river branches out. For around 70 km, until the Vietnamese border, it runs alongside the Mekong mainstream across Kandal province. In the wet season, the floodplains are submerged under several meters of water except for narrow strips of land along the main river levees. When the water recedes, a mosaic landscape made of small cultivated fields progressively emerge.

Central to this unique, multi-use ecosystem are preks-earthen channels 10 to 50 m wide, several meters deep, and a few kilometers long-that run perpendicular from the main rivers (Mekong and Bassac). They 'end' in low-lying areas known as boeungs, which sustain productive small-scale capture fishery during the flood season and rice farming during the dry season. Close to the main river levees, farmers cultivate fruit trees, sugarcane, beans and vegetables in small fields called chamkar that are seldom flooded.

In the 1990s, this mosaic landscape was rediscovered as one of the ''most productive farming system [. . .] in the whole of Cambodia" (JICA, 1998: 58). Our research showed that development projects that had been implemented in the area almost exclusively centered on increasing agricultural outputs by excavating preks that had silted up and by building sluice gates to enhance water control [START_REF] Venot | A multiplicity of prek(s): Enacting a socionatural mosaic in the Cambodian upper Mekong delta[END_REF]. People living by the preks had hardly been involved, and rather complex engineering interventions had been carried out based on sparse hydrological knowledge. In some cases, infrastructures collapsed almost as soon as they were built.

As part of our research, we wanted to challenge and expand this narrow focus.

From a livelihood perspective, we worried that agricultural intensification would mainly benefit wealthier (chamkar) farmers and adversely affect smallholders and capture fishermen. In line with social and ecological critiques of hard infrastructure solutions for delta management [START_REF] Wesselink | Trends in flood risk management in deltas around the world: Are we going 'soft[END_REF]; WWAP (United Nations World Water Assessment Programme)/UN-Water, 2018), we also wanted to explore 'nature-based' solutions able to sustain multiple more-than-human worlds and practices. For these reasons, our preferred unit of analysis was landscape transformation rather than single preks.

Against this background, we imagined that playing serious participatory games would allow us to learn about the many relations that make up the mosaic landscape. Game sessions could be imagined as ''hybrid forums" [START_REF] Callon | Acting in an uncertain world: An essay on technical democracy[END_REF] for collective articulation of alternative possibilities to landscape transformations, which might help to push development interventions in more sustainable directions.

Accordingly, we hoped to involve a range of 'local people' impacted by on-going prek rehabilitation projects. But as foreigners and newcomers we knew little about the intricacies of their affairs, conflicts, and aspirations, or how they shaped existing relations along the preks, not to mention with diverse outsiders to the mosaic landscape, like politicians, businesses, NGOs, and development agencies. Conversely, the 'locals' were similarly ignorant of our quite heterogeneous relations. Among those were the difference in orientation between AFD's technical teams, mainly engineers, who had been hired to rehabilitate the preks, and our research team, a mixed group of university-based researchers working on water and infrastructure issues. 7 These brief descriptions only touch the surface of those complex relations. Even so, they should make clear that the contexts in which the serious games were designed and played are inadequately understood with reference to general dichotomies such as 'locals ' and 'outsiders,' 'Cambodians' and 'foreigners,' or 'experts' and 'participants.' Instead, they took place in a context of numerous permeable boundaries and shifting alignments of interests, including our interest in influencing AFD's agenda for prek rehabilitation.

Setting (generous) constraints

The development of serious games was an iterative process. Starting in 2018, researchers from two French research institutes were involved in five 'game sessions,' each lasting one or two days. Before each session, objectives, rules, and tools were collectively defined with Cambodian partners from RUA and ISC in 'preparatory workshops' lasting three to five days.

The games, which differed between sessions, were played several times and modified with diverse participants including farmers-cum-fishermen, village chiefs, commune officials, ministerial staff, district and provincial administrations officials, staffers from NGOs and development agencies, and researchers (See Fig. 2 for a timeline).

The transformative component of the research project entailed numerous practical engagements with actors who had a role in shaping the mosaic landscape. In particular, there was a significant overlap between our research topics-water management, agriculture, infrastructure, and landscape transformation-and AFD's practical efforts to rehabilitate the preks. Accordingly, we interacted frequently with the WASP and WAT4CAM technical teams and AFD officials and we invited them as participants to game sessions from early on.

By the summer of 2020, the situation was changing. The WASP project had come to an end and WAT4CAM was still at an early stage. In the gap between these two projects, we perceived a margin of maneuver for influencing the new project to take a broader view of landscape rehabilitation. One way to do so might be to directly involve the new technical experts in serious game design and play. With skill and some luck, perhaps scenarios from the game sessions could lead to a broader interest in the landscape. The email exchange with which we began initiated this attempt and led us to co-design a game session with AFD 'experts.'

The games were played on a 'board' (e.g. a mosaic of colored or plain plywood 'tiles' -Figure 1-or a large plastic tarpaulin -top left panel of Figure 3), which represented different types of land located alongside a single prek or by a network of preks. Vignettes represented the most common crops found in the area (mango, vegetables, rice, etc. -top middle panel of Fig. 3) and action cards represented 'interventions' (infrastructure construction, excavation of preks, training of farmers, etc. -see top right panel of Fig. 3).

In the games, players assumed different roles, like 'farmer' or 'official.' They were assigned different 'loose' objectives relating to e.g. increasing agricultural or fishery production or ensuring the 'satisfaction' of the population (for an example, see the bottom left panel of Figure 3). These objectives could be achieved by placing vignettes and cards on the board and thereby modifying the prek system we had set up. Facilitators had pre-identified crops and possible interventions, but blank cards made it possible for players to add new elements. Some rules varied between game sessions, for example to explore differences between individual and collective decisions (which we assumed would facilitate the emergence of more equitable and environmentally sustainable future scenarios for the prek mosaic).

Behind the game play, a range of decisions about the effects of various 'events' (e.g. floods, droughts) and interventions (e.g. construction of infrastructure) on indicators like agricultural production, fish population, or 'satisfaction' had been ''rendered technical" in so-called calibrations. The purpose of these calibrations was not to quantify the actual impact of e.g. building a sluice gate, which would in any case be impossible. Instead, they reflected qualitative knowledge and broad estimates based on our research. Thus, building a sluice gate might lead to a 3 unit increase of agricultural production in the chamkar and a 2 unit increase in the boeung (due to improved water control), while decreasing fish population by 2 units (due to impeded water flows) (for the effects of building a sluice gate, see the top right panel of Fig. 3). 8Indicators were also adjusted for flooding. For example, if an 'extreme flood' occurred where a sluice gate had been built, chamkar production would decrease by 5 units rather than increase by 3. Such numbers strung together an array of assumptions. First, it was assumed that the building of a sluice gate would provide farmers with a sense of security and lead to intensified agriculture. Hence, the 3 unit increase during a normal flood. However, if a flood dam-7 Among the researchers there were further differences. Some were experts in Companion Modeling, and we adopted the approach due to their expertise in light of the transformative ambition of the DoUbT research project. For others, ComMod was not a central research identity, and the approach offered simply an interesting avenue of exploration. aged the infrastructure, the additional crops and investment would also be destroyed. Hence, minus 5 units. Again, this was not based on (non-existing) 'hard' data but on estimates of approximate consequences. After vignettes and action cards were placed on the board, floods or droughts were simulated by rolling a 12-sided dice, high numbers representing extreme floods and low numbers droughts. Once again, these probabilities were not based on hydrological modeling but rather on our 'sense' that the thresholds we used could generate a diversity of situations that would be interesting to explore collectively.

As this suggests, the game calibrations were quite 'loose.' But they were not 'neutral.' Since a major reason for playing the games was to enable collective speculation about possible landscape transformations in counter-point to hard infrastructure solutions, they pushed in that direction. In the case of extreme flood events, for example, players who had invested in dikes and sluice gates would suffer relatively higher losses than those who had opted for training courses or support for collective organization. The point was not to force players to choose those options but to make them appear somewhat more viable in a context where development is usually associated with the construction of cement structures.

The next section examines how constraints were negotiated at the level of board design. Rather than a process of depoliticization via technical simplification, this took the form of a series of disagreements about how and in which form to inscribe politics on the board. We then describe the entry of some unsurprising social and gender-related games of power. However, alongside these games of power, we glimpse other potentials. With the aim to turn serious games into technologies of humility it is crucial to learn from such glimpses. They are signs of the conditions under which participatory formats might do ''a little more than oppress the user" (Gomart, 2002: 517).

Designing the board

Let us examine the considerations that went into designing the game board with the WAT4CAM technical experts. It began with their insistence that the board had to properly represent 'their' project area (e.g. the preks they planned to rehabilitate). Otherwise, the experts argued, the game would not be serious.

This ran counter to the central assumptions of Companion Modeling in two ways. First, it would minimize the distance between game and reality and thus make it easier for real power relations to take over the games. Second, the familiar looking board would make it harder to imagine radical alternatives. Since we precisely wanted to explore alternative forms of landscape transformation there was a problem.

However, the demand for geographical realism could not be ignored. It was, after all, us who had pushed the skeptical technical teams to join us as designers. It was also us who hoped that the game would bring them face to face with ideas sufficiently intriguing to bring into their plans for prek rehabilitation. Thus, we had to come up with a design that was 'real' enough for our partners to accept as 'serious.' But it still also had to support our own agenda.

We tried to solve this issue in two ways. The board had to resemble the project area 'enough' but what that meant was open to interpretation. As a result of previous interactions, we had made some inroads with the argument that considering several adjacent preks together made better sense than a focus on single channels: the WAT4CAM technical teams had begun to organize work around clusters of preks. Accordingly, we suggested that the board could represent the first cluster proposed for rehabilitation (see Fig. 1). But we insisted that the board should encompass the whole boeung in which this cluster 'terminated'. The technical experts knew quite well that this made hydrological and agricultural sense. 9 Even so, it spelled trouble since only part of the boeung fell within their official 'project area,' which somewhat arbitrarily defined an old Khmer rouge canal running through the middle of it as its western limit. It would be difficult to make interventions outside the formal 'project area' but also to dismiss requests made during a participatory workshop organized in relation to the project. Ironically, this meant that geographical realism was a generous constraint for the technical teams but only halfway through the boeung after which it immediately became ungenerous. In contrast, including the whole boeung was important to us. It highlighted the legitimacy of farmers' concerns beyond the confines of the project -turning its full inclusion into a generous constraint.

These negotiations exhibit tensions between our research agenda and the views of AFD's technical teams. They also challenge the conventional wisdom that ''rendering technical" and 'simplifying' is always depoliticizing. Because the disagreements had nothing to do with being for or against constraints in general. They were not about whether or not politics should be embedded in the board design but about the particular form of politics that different kinds of constraints would inscribe in game sessions. The eventual design was a compromise that tried to maintain relative generosity despite significant disagreements. The board was connected to a specific project area-the first cluster of preks considered for rehabilitation-but it also connected preks considered for rehabilitation to previously rehabilitated ones and it covered the entire boeung.

The fact that a particular 'politics' had been inscribed on the board did not mean that extant development priorities now silently dominated the whole scene. To the contrary, the relative importance of the board, the rules, and the calibrations all varied with other constraints introduced by participants in the course of playing. These local, emerging constraints were often so forceful that we had difficulties maintaining our focus on alternatives for landscape transformation.

Games of power

As a consequence of the permeable boundary between game sessions and real world, all kinds of problems, conflicts and grudges seeped into the games. 10 Tackling them required an enormous amount of time and energy.

If we nevertheless keep the discussion brief, it is because power relations in development contexts are so well studied (e.g. [START_REF] Cornwall | Whose voices? Whose choices? Reflections on gender and participatory Development[END_REF][START_REF] Lund | Revisiting the issue of elite capture of participatory initiatives[END_REF][START_REF] Ribot | Recentralizing while decentralizing: How national governments reappropriate forest resources[END_REF] that their primary relevance here is as a counter-point. They exhibit what we constantly struggled against to provide the games with a chance to be something a bit more generous as well.

The following provides an example: 11

Participants sit by a board designed by a group of farmers. The discussion focuses on which canals should be excavated in the boeung. I ask participants to move tokens on the board to reflect the discussions so it is easier to keep track of things. Behind a village chief, three women are quietly discussing. In the previous session, they had been quite vocal about which canals they wanted excavated, yet they had put tokens on the board only reluctantly, and with direct encouragement. The male participants on the other side of the table showed no such compunctions. They quickly placed their tokens and cards to show what they wanted.

Concerned that the women's viewpoint is about to be sidelined again, I discreetly pass a token and nod towards the table. Nobody moves. Uncertain, I ask the facilitator to make everybody show their remaining tokens. Slowly, the women raise their hands. But as I invite them to step forward, they gently touch the shoulder of the chief sitting in front, and pass their tokens on (Observation notes, game session, August 2020, JPV)

We are confronted with social constraints relating to identities, gender, and power, loaded into the game by the players themselves. The image of the women passing the token to the village chief makes clear that defining something as a game is insufficient to create a space for open-ended speculation. Evidently, the critiques of participation are not irrelevant.

They are also pertinent with reference to the following, where a man refrains from acting within the game because the 'same act' would cause conflict in reality: This session featured two groups discussing what to do around a board they had designed to represent the same prek. One group wanted to build a sluice gate but decided not to do it because they knew that the village chief, who was in the other group, would object. 12 Three women defer to the village chief. A man declines to act in a way that would mobilize a real conflict. Both examples show players busily closing the very gap between game and reality, which ComMod argues is prerequisite for keeping power relations at bay.

We are thus faced with a question. What happens when the two core methodological tenets-a healthy distance between game and reality must be maintained and players should be allowed to modify the design of the game-conflict? Clearly, the invitation to redesign the game can also work against aspirations to facilitate collective explorations.

The following section starts from this tension. It shows that although 'distance' does not really prevent power relations from influencing the games, the critiques of participation are still only partially vindicated. Because as participants kept tinkering with constraints, glimpses of other landscapes and relational possibilities also emerged. 9 Hydrologically, the preks are interconnected 'through' the boeungs, which are thus crucial for water control and management. In agricultural terms, it is often the same farmers who cultivate the boeung and chamkar; they adopt complementary farming strategies for these different types of land.

10 Permeability is recognized by both critics of participation and participatory modelers. For critics, the boundary is so permeable as to be a fiction, which is why power relations always contaminate participatory formats. For the modelers, the point of distance is to make the boundary temporarily less permeable to keep those power relations out. In fact, distance does not suffice for this job. As we discuss below, what nevertheless makes it possible for games to be something more than mere games of power is tinkering with patterns of constraints.

11 All descriptions are based on short-hand observation notes and dialogue excerpts (some translated 'live' from Khmer to English by a Cambodian facilitator). They have been grammatically corrected.

12 Discussions during the workshop revealed that the village chief ran a small pumping station and sold water to farmers. Building a sluice gate entailed-at least theoretically-year-round water availability in the prek, which might ruin this business.

Jean

Mosaic glimpses

To safeguard the methodologically required 'distance' between games and reality, by far the majority of games were played on boards that randomly assembled land types 13 and connections between rivers, preks, and canals. However, this perplexed many players, who failed to see the relation between our 'serious questions' and generic boards that didn't resemble the landscapes they knew. Similar to the technical experts, though for different reasons, most local people would also rather play on boards they could associate with specific places. Which, again, is precisely what we wanted to avoid.

The following exchange followed the presentation of one 'random' board prepared for a session. The official immediately wants to connect the board to a known place. In fact, this was easily done, since it merely required the players to agree that the board showed the right bank of the Mekong. As the game continued on that basis, the distance between game and reality did not really collapse-after all the physical features of the board did not change-but it was also not maintained, since everybody now associated the game with a specific area of the mosaic landscape.

However, in this case, rather than a predictable 'game of power' the result was resonance between researchers and players at the level of game orientation. The spontaneous decision to have the board represent a good chunk of the mosaic landscape created speculative leeway. In contrast with monotone technical discussions centering on the depth of excavation of a specific prek, or whether or not to build a pumping station or a sluice gate [START_REF] Venot | A multiplicity of prek(s): Enacting a socionatural mosaic in the Cambodian upper Mekong delta[END_REF], quite varied scenarios began to emerge. Some focused on intensifying vegetable and fruit produce in the chamkar and others on protecting boeungs from extreme flooding. Some concentrated water infrastructures in specific 'hot-spots,' while others distributed structures over the entire landscape. There were mosaic glimpses of other landscapes.

Meanwhile, other sessions exhibited permeable boundaries between types of players, or between players and researchers.

After forming groups, farmers gather around a board in the pagoda. The facilitator begins to explain what the board shows, assigning names to the preks and canals. After a moment, farmers start discussing, asking questions, and adding pieces of wood on the board. They place different market places and a pagoda. After more discussion, they add blue tiles for a missing canal. In the afternoon, officials are asked to look at the board designed by farmers and vice versa. Looking at the number of canals shown in the boeung on the 'farmer board', a commune chief is surprised to count four. He asserts there should only be three canals [as in the board we had predesigned, which was modified by the farmers], and begins to remove the blue tiles. But a village chief stops him and confirms: ''there is a canal here" and, pointing to another area of the board adds ''and here too." After discussing, they further modify our board design by shifting around yet another canal. (Observation notes, game session, August 2020, JPV).

Here we see different player orientations to the 'power relations' inscribed in the game. Officials did not think to question the board, which had been designed in accordance with our 'expert knowledge.' They took it for granted and simply played on it. However, farmers redesigned the board by adding a missing canal to fix our mistake. Noticing the discrepancy between the boards, the commune chief was thus startled. But with further confirmation that the added canal did indeed exist, came an awareness that our 'expert' board might have other flaws -which led to curiosity, dialogue, and further modifications.

A final excerpt illustrates provisional changes in power relations. Here, the players improvised a consensual result in direct contravention of game rules.

The two groups (farmers and officials) were asked to separately negotiate a set of interventions on their boards. Quickly, officials from group 1 went to see the farmers in group 2, asking about what they had done on the board, then returning to replicate it on theirs. Farmers went in the other direction and further modified the position of the tokens. We let it slide. . . (Observation notes, game session, August 2020, JPV) This session had, in fact, begun with quite stern rules against interactions between the two groups. They were simply ignored. As people moved back and forth and communicated, the boards started to look more and more similar. For example, small, identical bridges first proposed by farmers appeared in the same area on both boards. This improvised harmony seemed to strengthen the proposal: the attendant WAT4CAM project engineer took copious notes and asked many follow-up questions.

From these 'glimpses' can be drawn both 'critical' and 'constructive' conclusions. We observe again that the simplifications and forms of ''rendering technical" specific to the games cannot be reduced to 'repression' or 'depoliticization.' Instead, tinkering to keep the constraints flexible in different situations supported problematic (imperfect and in friction) articulations of different forms of infrastructural politics in relation to the preks. That tinkering was continuously required to keep the games on track makes clear that 'distance' is hardly a methodological solvent for issues of power. Conversely, the fact that relations between different participants, and researchers did transform in the course of playing also indicates that boundaries can be kept relatively permeable by attentive experimentation with generous constraints.

Calibrating participation

We have mainly concentrated on the game sessions. This surface level focus has facilitated examination of the interplay between many kinds of constraints. But it leaves other aspects untouched. Central are the calibrations, which, because they are at the technical heart of the games, are prime suspects for secretly depoliticizing the issues. For the same reason, they were rarely subject to explicit discussion in game sessions although we always mentioned them (Fig. 3).

However, calibrations were always discussed backstage, during the preparatory workshops. Here, for example, JPV has asked whether our partners understood the game 'mechanics': As we see it, this perplexity indexes two related issues. First, the seriousness of the games is once again connected to the accuracy or 'reality' of numbers. Second, more subtly, but more fundamentally, our admission that the games were not based on 'hard facts' also ran counter to the conventions of development workshops, where facilitators and experts are supposed to convey authoritative knowledge to participants imagined to lack it. 14 In light of this expectation, our vague designs and qualitative calibrations often confounded our partners from ISC and RUA. When they got their own opportunity to design a game, the approach they took was strikingly different.

After years of working together, it was no longer a surprise that we had rather different understandings of the serious games. However, the magnitude of those differences only crystallized towards the end of our first project. Around this time, our primary concern was to ensure that our partners could facilitate participatory game sessions on their own and they piloted the last preparatory workshop we held. Not limited by our focus on water and infrastructure, the next game would center on safe agricultural product value chains, a topic they had been interested in since the beginning our collaboration.

Our partners completely dissolved the game design. In this new game, there were no preks, land-types, floods and droughts, sluice gates, or even a board. Instead, each player sat in front of a stack of wooden tiles, which they could assemble to represent individual farms. They placed crop vignettes on tiles, purchased seeds and inputs, and sold products with numbered tokens and bills representing income. 15 As the board decomposed into a set of movable elements, all constraints seemed to vanish. The game was now much more generic than we had imagined, but in a totally different way.

And yet, behind the exploded game surface, our partners introduced very rigid calibration measures. They wanted to account for the exact price of a bag of pesticides, the correct market value of fruits and vegetables, and precisely how many extra crops a farmer would gain per added bag of inputs. The demand for realism was moving from the board surface into the black box of calibrations.

The official rationale was to make the game more easily relatable to famers. But when we quizzed Vira and Kong about the hours they spent concocting numbers we were told quite a different story. Realism, they now insisted, was crucial to convince famers ''that safe agricultural production is better than current practices." The farmers had to be made to understand the importance of ''coordinating to meet the demand of potential buyers." As sessions were redefined as behavioral scripts and guidelines for the players, 16 the game itself changed into a tool for raising awareness about topics and solutions defined in advance by the experts -not the foreign research experts, mind you, or even AFD's technical experts, but Cambodian NGO experts and university staff.

From a broader perspective, we would suggest that these divergent strategies of calibration relate to different orientations to the making of development knowledge in general. On one side is the hubristic idea that for games to be serious, the major empirical components (and preferred outcomes) must be known in advance. In our opinion, this is neither very generous nor serious.

In contrast, taking uncertainty with respect to the system, its relations, and its possibilities seriously pulls serious games in the direction of hybrid forums that facilitate open-ended explorations. They are provided with a chance to become technologies of humility. We have thus come full circle.

Conclusion

This paper has used a series of serious games played in Kandal, Cambodia as an entry point to reexamine relations between development projects, participatory formats, and sustainable futures. It is undeniable that serious games simplify complex problems and render them technical. But, entering the black-box, we have shown that the results are not reducible to a matter of domination. What is required for the games to do 'a little more' than repress the players is keeping a range of socio-technical patterns of constraints relatively generous.

In Kandal's mosaic landscape, the 'little more' we aimed for related to two issues. In contrast with prek rehabilitation that focused on intensifying agriculture by excavating channels and building water control infrastructure, we were inspired by recent alternatives to delta management centering on 'nature-based solutions.' We also wanted to learn from a variety of voices-not just experts and policy-makers but also farmers and fishermen-what they thought mattered in and about the landscape. Moreover, as time passed, we began to wonder how those voices might be heard by the technical teams tasked with the next rehabilitation project. Thus, we came to envision serious games as hybrid forums for the collective exploration of, and speculation about, alternative landscape scenarios that might sustain a variety of more-than-human worlds.

As discussed, this involved many forms of ''rendering technical" and numerous constraints introduced by the researchers and their partners, from calibrations and board design to rules for player interactions, and self-imposed player sanctions. None of those constraints were neutral. As we have shown, however, they are also not well captured by general rubrics like repression or depoliticization. To understand the divergent politics of participatory formats like serious games, it is crucial to zoom in on the specificity of constraints.

Constraints can be more or less supple and flexible, or rigid and fixed, in different aspects. For example, Companion Modeling promotes 'distance' as a generous constraint that is meant to support collective experimentation by reducing the expression of real power relations in the games. But distance cannot really keep such relations away, and both technical experts and local villagers tended to view games that facilitated actionable propositions as more generous.

What can prevent games from becoming mere contests of power is ongoing tinkering to keep the patterns of constraints generous and the boundaries between virtual and real worlds, insiders and outsiders, and present and futures relatively permeable.

In our view, the most generous achievement of the games was broadening the exploration of sustainable futures to the level of the mosaic landscape. Most scenarios retained a focus on infrastructure development but they were strikingly heterogeneous 14 These conventions may also explain why calibrations were hardly discussed during game sessions. 15 This game was played with farmers in February and March 2020 without our involvement (Figure 2). 16 The point, again, is not that only these calibrations constrained the players. It is that the manner of constraining differed. While our calibrations made options other than hard infrastructure slightly more viable in the face of extreme events, the new game had a single winning scenario premised on avoidance of chemical inputs. At the same time, however, our partners' decomposition of the game also created generative openings. For example, it made us realize the boards could be redesigned on the spot. This came in handy when we had to make the board represent the WAT4CAM project area 'realistically.' Jean-Philippe Venot, Casper Bruun Jensen, E. Delay et al. World Development 151 (2022) 105779 compared with monotonous debates about excavation options for single preks.

Although the new technical teams continue to harbor doubts about 'seriousness,' they now intend to involve local actors in elaborating their 'master plan' for future prek rehabilitation. The small bridges spontaneously emerging on separate game boards have been translated into engineering blueprints. And we have learned from our partners at RUA and ISC that speculative environmental scenarios need to go beyond our obsession with water issues. These are all mosaic glimpses of how participatory formats can do a little more than oppress the user.

They can operate as experimental systems for collective exploration, which are serious to the extent that they work like technologies of humility rather than hubris.

Humility must be understood in a double sense. It acknowledges uncertainty and a need for collective learning premised on appreciation of heterogeneous voices. In this sense, it involves an effort to push away from the embedded epistemic hierarchies of development. As hybrid forums, serious games support efforts to 'do difference together' [START_REF] Verran | Doing difference together: Towards a dialogue with aboriginal knowledge authorities through an Australian comparative empirical philosophical inquiry[END_REF] in explorations of more-than-human worlds and divergent practical ontologies. At best, they can begin to redirect active-passive relations between development 'experts' and 'recipients' towards more reciprocal constructions.

At the same time, humility entails awareness that participatory games rarely have world-shattering consequences. Most things remain in place, and many of them-from the practices of French development funders and the orientations of water engineers to Cambodian political hierarchies and embedded gender inequalities-are very hard to change however much we would like to. From this angle, humility evokes a pragmatics of aspiration in which smaller changes count for something too. The bridges, for example, are a small success for people living along these preks. But they are also devices that bridge-or scale-between that achievement and our landscape approach, which they make appear slightly more viable.

In turn, we have tried to make the notion that more-thanhuman mosaic landscapes can be supported by experimenting with serious games as technologies of humility slightly more viable in these pages. Rather than merely depoliticizing, experimental participatory formats can help articulate the multiple practical ontologies involved in prek rehabilitation. Learning from this multiplicity matters for moving towards sustainable forms of living in Kandal and elsewhere.

2

  The ''Water Resources Management & Agricultural Transition for Cambodia Project" (WAT4CAM) is a 200 Million Euros, six-year project (2019-2025), co-funded by the French Agency for Development (AFD) and the European Union (EU) and implemented by the Cambodian Ministry of Water Resources and Meteorology (MoWRAM) and the Ministry of Agriculture, Forestry and Fisheries (MAFF) in five provinces. It builds on the earlier Water and Agricultural Sector Project

Fig. 1 .

 1 Fig. 1. A prek in the mosaic landscape of in Kandal (left) and discussion around a serious game, Kandal province, August 2020 (right) (The Authors).

Fig. 2 .

 2 Fig. 2. Research and Engagement: Timelines of game sessions.(See below-mentioned references for further information.)

  JPV: Do you want to put a gate there? Participant: Yes, that is what we are discussing JPV: Here [passing a token], you can put it if you want Participant: No. . . It is ok JPV: I don't understand, if you think it is a good idea, why don't you put it? Participant: Before we put it, we need to ask [points to a man sitting in the other group] JPV: Ah ok, why don't you go there and ask him then Participant No. . . It is ok (Dialogue with farmer, game session, May 2019)

  Official: What places does the board show? JPV: It's pretty much up to you. Official: We need to know, if you are asking what we think should be done. JPV: We can say that this [pointing to a blue area on the side of the board] is the Mekong . . .or the Bassac, whichever you want. Official: It is the Mekong because. . . [Points to a light green patch on the board] JPV: [All seem to agree] Let's say it is the Mekong then. Official: Ok, we can go ahead now. (Dialogue with official, game session, May 2019)

  -Philippe Venot, Casper Bruun Jensen, E. Delay et al. World Development 151 (2022) 105779

  Vira Oum: What I still do not understand is the calibration -How does it work? Why did you choose the numbers you chose? JPV: The number are not what is important. It's the overall logic that matters. 13 Defined in terms of likelihood of flooding and types of agriculture. Jean-Philippe Venot, Casper Bruun Jensen, E. Delay et al. World Development 151 (2022) 105779 Kong Sinh: Me too, I still do not get it, what do the numbers represent?JPV: The numbers do not matter. We could have chosen 2, 4, 6, it does not matter. What matters is that they make sense in relation to each other and that it is easy enough to manipulate in the game and does not take too much time.

	Vira Oum: Mhhhh. . .
	(Preparatory workshop, May 2019)

In line with the observation that: ''the concept of the boundary is one of the least subtle in the social science repertoire"(Strathern 1996: 520).Jean-PhilippeVenot, Casper Bruun Jensen, E. Delay et al. World Development 151 (2022) 105779

The game 'units' did not have any equivalent in the 'real' world (such as tons, bags) but were meant to represent concepts such as production, income, etc. How a specific intervention impacted each indicator was shown on the corresponding action card (Figure3) and materialized by tokens on the board.

Acknowledgments

This research has been conducted in the framework of the Deltas' Dealings with Uncertainty (DoUbT) and COSTEA projects. Financial support from the French National Agency for Research (ANR) and the Japan Society for the Promotion of Science (JPSP) in the context of the Open Research Area (ORA) for the Social Sciences program, and from the French Agency of Development (AFD) through the Association Française de l'Eau, l'Irrigation et le Drainage (AFEID) is duly acknowledged. The authors would like to thank their colleagues from the Irrigation Service Center and the Royal University of Agriculture as well as all participants who contributed to designing and played the serious games described.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.