Capturing an Escaping Prisoner? (Something I thought about in my car) So, I was in my car listening to the radio when I was listening to a story about the captured New York prison escapees, and it had me thinking:
In a hypothetical large fixed area of land, let us present a prisoner p and a cop, c. Assuming both move randomly, would a prisoner, on average, be caught quicker if they remained idle in one spot, or kept on moving? 
I've taken coursework in mathematical modeling of epidemiology, and I would assume that the prisoner would be caught faster if both the cop and the prisoner were moving based on the following:
in mathematical modeling, contact rates are simplified via a simplified version of the "mass action" principle from chemistry, which states that a contact rate can be defined by the products of two concentrations. If we assume movement is a "concentration", it would imply that if a escaped prisoner and a cop were both moving, that the overall order of the reaction rate be 2nd order, as opposed to a first order system where it was just the cop moving. 
Just curious, is there a probabilistic approach to this that would strengthen this argument further or maybe even give a completely different answer? Would it be dependent on their relative rates? 
 A: I did a simulation of this, assuming that the two players move randomly within the area. 
    ## given a single coordinate of a single player, return a new position by a valid move
moveCoord = function( playerCoord, boardSizeCoord){
    move = sample(c(-1,0,1), 1)
    newCoord = playerCoord + move
    if (newCoord < 1 || newCoord > boardSizeCoord)
        newCoord = moveCoord(playerCoord, boardSizeCoord)
    return(newCoord)
}

## given a player's location, return a new location by a valid move
movePlayer = function(playerCoord, boardSize){
    newX = moveCoord(playerCoord[1], boardSize[1])
    newY = moveCoord(playerCoord[2], boardSize[2])
    newCoord = c(newX, newY)
}

n = 10 ## board is nxn
boardSize = c(n,n) ## board size
steps = 1000000 ## how many moves to try in each run of simulation, before they fail 
copCoord <<- c(floor(n/3), floor(n/3)) ## cop is at the first third vertically and horizontally (rounding down)
prisCoord <<- c(ceiling(2*n/3), ceiling(2*n/3)) ## prisoner is at the second third vertically and horizontally (rounding up)

## how many steps until both players intersect?
simulate = function(prisMove = FALSE){
    for( i in 1:steps){
        copCoordMat[i,] = copCoord
        if(identical(copCoord,  prisCoord)){
           ## print(paste("Caught at ", i))
            return(i)
            break
        }
        else {
            copCoord = movePlayer(copCoord, boardSize)
            if(prisMove == TRUE)
                prisCoord = movePlayer(prisCoord, boardSize)
            ##print(copCoord)
          }
    }
}

reps = 1000
noMoveVec = replicate(reps, simulate(prisMove = FALSE))
yesMoveVec = replicate(reps, simulate(prisMove = TRUE))
summary(noMoveVec)
summary(yesMoveVec)

Here are the results. In noMoveVec, only the cop moves; in yesMoveVec, both players move:
   > summary(noMoveVec)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    7.0    71.0   131.0   176.6   231.0  1056.0 
> summary(yesMoveVec)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    5.0    56.0   115.0   160.0   216.2  1123.0 

In 1000 runs of the simulations on a 10x10 grid, the moving prisoner got caught a little faster on average. I wasn't sure how much of the effect was due to the initial positions. So I tried it again having the cop in the bottom left corner and the prisoner in the top right corner. There's a way bigger effect in this case.
summary(noMoveVec)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   25.0   195.8   410.5   561.2   758.2  3150.0 
> summary(yesMoveVec)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   15.0    70.0   126.5   173.2   233.2  1057.0 

