globals [ Current-Agent Competitors ] breed [agents agent] agents-own [ current-value param-x param-y wins ] to setup clear-all create-agents 1 [ set param-x current-x set param-y current-y set current-agent self ] create-agents number-of-competitors [ set param-x competitor-x set param-y competitor-y ] set competitors agents with [self != current-agent] setup-plot end to setup-plot set-current-plot "Trade Off" set-plot-x-range min-x max-x set-plot-y-range min-y max-y plot-pen-up plotxy max-x 0 plot-pen-down plotxy min-x 0 plot-pen-up end to reset-agents ask current-agent [ set param-x current-x set param-y current-y set wins 0 ] ask competitors [ set param-x competitor-x set param-y competitor-y set wins 0 ] end to simulate-competitions let winner nobody ask agents [set wins 0] repeat number-of-iterations [ ask agents [ ; set current-value random-normal param-y param-x ; x is standard deviation set current-value random-normal param-y (sqrt param-x) ; x is variance ] set winner max-one-of agents [current-value] ask winner [set wins wins + 1] ] end to estimate-y reset-agents ask current-agent [ set param-y min-y simulate-competitions let low-y-value (wins / number-of-iterations) set param-y max-y simulate-competitions let high-y-value (wins / number-of-iterations) y-search-step min-y max-y low-y-value high-y-value 0 set current-y param-y ] end to y-search-step [low-y high-y low-y-value high-y-value cur-iteration] if cur-iteration = estimate-iterations [ stop ] let mid-y low-y + ((high-y - low-y) / 2) set param-y mid-y simulate-competitions let mid-y-value (wins / number-of-iterations) ifelse mid-y-value >= (1 / (1 + number-of-competitors)) [ y-search-step low-y mid-y low-y-value mid-y-value (cur-iteration + 1) ] [ y-search-step mid-y high-y mid-y-value high-y-value (cur-iteration + 1) ] end to batch setup let cur-x min-x while [cur-x < (max-x + stepsize-x)] [ set current-x cur-x estimate-y update-plot set cur-x cur-x + stepsize-x ] end to update-plot set-current-plot "Trade Off" set-plot-x-range min-x max-x set-plot-y-range min-y max-y plotxy current-x current-y plot-pen-down end @#$#@#$#@ GRAPHICS-WINDOW 9 322 181 515 16 16 4.91 1 10 1 1 1 0 1 1 1 -16 16 -16 16 0 0 1 ticks INPUTBOX 8 65 163 125 Number-Of-Competitors 10 1 0 Number INPUTBOX 8 189 163 249 Competitor-x 1 1 0 Number INPUTBOX 8 127 163 187 Competitor-y 0 1 0 Number INPUTBOX 8 250 163 310 Number-Of-Iterations 5000 1 0 Number INPUTBOX 169 127 324 187 Current-x 2.000000000000001 1 0 Number INPUTBOX 169 65 324 125 Current-y -0.5234375 1 0 Number INPUTBOX 513 65 668 125 Min-x 0 1 0 Number INPUTBOX 513 127 668 187 Max-x 2 1 0 Number INPUTBOX 350 65 505 125 Min-y -4 1 0 Number INPUTBOX 350 127 505 187 Max-y 4 1 0 Number INPUTBOX 350 190 505 250 Estimate-Iterations 10 1 0 Number INPUTBOX 513 190 668 250 Stepsize-x 0.05 1 0 Number MONITOR 513 254 580 299 # x Steps 1 + ((max-x - min-x) / stepsize-x) 3 1 11 MONITOR 171 266 297 311 Est. Probability Wins ([wins] of current-agent ) / number-of-iterations 3 1 11 BUTTON 171 191 235 224 Setup setup NIL 1 T OBSERVER NIL NIL NIL NIL BUTTON 171 230 311 263 Estimate Probability reset-agents\nsimulate-competitions NIL 1 T OBSERVER NIL NIL NIL NIL BUTTON 350 263 439 296 Estimate y Estimate-y NIL 1 T OBSERVER NIL NIL NIL NIL PLOT 678 65 996 380 Trade Off Variance Mean 0.0 1.0 -1.0 1.0 true false PENS "default" 1.0 0 -16777216 true BUTTON 586 254 649 287 Batch batch NIL 1 T OBSERVER NIL NIL NIL NIL TEXTBOX 14 10 458 32 Expected and variable performance: a trade-off 16 0.0 1 TEXTBOX 13 42 163 60 After James March (1991) 11 0.0 1 TEXTBOX 351 48 503 76 Estimate y for given Current-x: 11 0.0 1 TEXTBOX 514 47 786 75 Plot trade-off line for range of values of Current-x: 11 0.0 1 TEXTBOX 171 50 321 68 Estimate chances of winning: 11 0.0 1 TEXTBOX 209 321 359 377 Agents perform by sampling from a Normal distribution with mean = y and standard-deviation = x.\n 11 0.0 1 @#$#@#$#@ WHAT IS IT? ----------- An illustration of how to trade-off mean performance and variance in performance when trying to come first in a competition. As described by James March (1991), figure 6. An agent (firm, player etc.) has N competitors in some competition for which performance is normally distributed. Competitors are homogeneous. They all draw their performances from N(0, 1), i.e. a normal distribution with mean = 0 and variance = 1. The one special agent also draws its performances from a normal distribution. If the given agent draws from the same distribution as its competitors, its chance of performing best, or coming first, will be 1 / (N + 1). To have better chance than its competitors, it could, of course, raise its expected (mean) performance. Less obviously, however, raising its variance in performance will also improve its chances of coming first. Indeed, if a change in behaviour affects both mean and variance, it might even raise its expected performance yet lower its chances of coming first! HOW IT WORKS ------------ A competition consists of all N + 1 agents drawing one performance value from their Normal distributions. The one winner is one of the agents with the largest performance value. If the competition is reiterated many times, the proportion of times a particular agent wins provides an estimate for that agent's chances of winning, given their current performance distribution. Different parameter values for that distribution give different chances of winning. HOW TO USE IT ------------- Choose the number of competitors. Click "Setup" to create all agents. The special agent uses the distribution N(current-y, current-x). Click "Simulate" to return an estimate of its chance of winning, calculated by running competitions for Number-Of-Iterations. For a given value of current-x (variance), click "Estimate y" to approximate the value of y (mean) at which the chances of winning are 1 / (N + 1). The search for a y value begins with max-y and min-y and recurses, halving the range each iteration for a given number, Estimate-Iterations. Click Batch to estimate y for values of x from Min-x to Max-x. These are plotted on the chart. THINGS TO NOTICE ---------------- Sampling from a Normal distribution (random-normal) is time consuming in NetLogo. Try the program with Number-Of-Competitors = 2 initially. Then try larger values. (1000 may take several minutes!) As the number of compeititors increases, the y-x trade-off line has sharper slope. If you see it kinking at the ends, you may need to increase Max-y and decrease Min-y so as to increase the range over which it searches for y estimates. THINGS TO TRY ------------- How many iterations do we need for a sufficiently accurate estimate of the chances of winning or the trade-off line? EXTENDING THE MODEL ------------------- What if you score a "win" for being in the top 10, or top 10%? What if performance follows some distribution other than Normal? RELATED MODELS -------------- March (1991) presented the trade-off in a paper describing a model of organisational learning. His point was that learning does not always confer competitive advantage. It can raise a firm's expected performance of some task, but by standardising more and raising reliability, it can also lower variance in performance. Its relative performance in the market depends on both expected performance and variability. Most papers on organisational or collective learning assume that raising expected performance is always desirable. CREDITS AND REFERENCES ---------------------- March, James G. (1991) "Exploration and Exploitation in Organizational Learning". Organization Science, Vol. 2, No. 1, Special Issue: Organizational Learning: Papers in Honor of (and by) James G. 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