# Penalty Kicks in Soccer

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## WHAT IS IT?

This is a model of penalty kicks in soccer. Being a zero-sum mixed-strategy game involving a finite number of players, it has a Nash equilibrium. The Nash equilibrium is a set of strategies chosen by each of the players which result in an optimal situation for all. For any player in equilibrium, deciding to change the proportion of any pure strategy used over another will result in having chosen a worse strategy than before. The agents in the model each try to employ different strategies in attempts to learn to play in that Nash equilibrium. By default, the Nash equilibrium exists at around [40%, 60%] for all players. The pure strategies involved are [L, R], non-natural or natural side. Note that the natural side for a right-footed player is his left and the keeper's right, indicated by R.

The analysis of this theory of minimax play in penalty kicks is put forth by Ignacio Palacios Huerta (2003).

## HOW IT WORKS

Goalkeepers learn adaptively. The algorithm used here is fully described in the Info Tab of the El Farol model in the models library.

Penalty takers learn goal-oriented. One takes a cheat sheet with information about his opponent's choices when he failed, and exploits that knowledge by choosing the side on which the opponent let in the larger amount of goals.

The length of the cheat sheet, and the amount of data available to an agent to create his cheat sheets is given by MEMORY-SIZE.

At any moment, either goalkeepers are learning or penalty takers are learning. This is given by STUDENTS. Teachers, on the other hand, have a slider TEACH-NAT-TENDENCY, which will set the probability of the teachers choosing their natural side over their non-natural side.

That's how the learning works, and each penalty kick itself is geared to replicate real-life data, so physically they work as they do in real life. Ball goes in, he scores!

## HOW TO USE IT

To use the model, set the NUM-TAKERS, MEMORY-SIZE, NUM-STRATEGIES, TEACH-NAT-TENDENCY, choose which breed to make learn, press SETUP, and then GO.

There many plots and monitors available to analyze the data. The monitor for scoring rates is aggregate, but the scoring rates on the plot is measured at each tick for the last NUM-TAKERS players. The plot and histograms show each breed's natural side tendencies, also measured for the last NUM-TAKERS players. The four monitors which report the isolated success rates according to footing can be used to compare results to the prediction of equal success rates among pure strategies.

## THINGS TO NOTICE

The 3D visual is awesome.

## THINGS TO TRY

The manual settings exist to extend the option of further analysis. Set MANUAL switch to ON if you wish to use these settings. The powers indicate the power of the ball when kicked to either side. For example, it might be useful to know what the total success rate is when all players choose their natural side 50% of the time.

Run the model with different settings for MEMORY-SIZE and NUM-STRATEGIES. What happens to the variability in the plots?

## EXTENDING THE MODEL

At the bottom of the code, there is a botched equation for calculating the Z-score of a sample list to know the probability of it being non-random. Try to get this right! There is a bit of code in the GO function as well to get you started on the implementation if you get the Z-score function right.

It is not too hard to change the learning algorithms for the players. Try adapting certain cognition codes from other models.

## NETLOGO FEATURES

This model does not run on ticks. A tick here represents a single penalty kick.

Lists are used to represent choices, results, and strategies.

n-values is useful for generating random lists.

Histograms provide an interesting visualization, but notice that the upper limit to each interval is open.

## RELATED MODELS

El Farol, Traffic Grid

## CREDITS AND REFERENCES

Palacios-Huerta, Ignacio (2003) http://www.palacios-huerta.com/docs/professionals.pdf

## Comments and Questions

breed [ keepers keeper ] breed [ takers taker ] keepers-own [ dive ;;how fast he dives (the taker doesn't move) choices choice results result strategies best-strategy tendency cheat-sheet1 cheat-sheet2 ] takers-own [ choices choice results best-strategy strategies tendency cheat-sheet1 cheat-sheet2 ] ;;same stuff turtles-own [ power ] ;;how fast the ball's moving globals [ ;;for statistics goals misses saves ;;for functions over? ;;true when we know the result of the kick gk ;;the keeper whose turn it is tk ;;the taker whose turn it is ball ;;the ball turtle ;;for plotting allresults gk-allchoices tk-allchoices gk-tendencies ;;list of each player's tendencies noted after his turn tk-tendencies ] ;SETUP FUNCTIONS to setup clear-all-plots reset-ticks setup-plots clr ask patches with [ pzcor = -4 and pycor = 1 ] [ if pcolor != green [ line-field set-goal ] ] ;;set up field only if it isn't already set setup-turtles end to clr ask turtles with [ color != white ] [ die ] ;;clear keepers, takers, and the ball, but not the goal set over? false set goals 0 set misses 0 set saves 0 set gk-allchoices [] set tk-tendencies [] set gk-tendencies [] set tk-allchoices [] set allresults [] end to line-field ask patches with [ pzcor = -4 ] [ set pcolor green ] ask patches with [ pycor = 36 or pycor = -18 or (pycor = 18 and abs pxcor <= 30) or (abs pxcor = 30 and 18 <= pycor and pycor <= 36) and pzcor = -4 ] [ set pcolor white ] ask patches with [ pycor = 0 and pxcor = 0 and pzcor = -4 ] [ set pcolor white ] ask patches with [ precision (distance patch 0 0 0) 0 = 30 and pycor <= -18 and pzcor = -4 ] [ set pcolor white ] end to set-goal ask patches with [ (pycor = 38 and abs pxcor <= 12) or (abs pxcor = 12 and 38 >= pycor and pycor >= 36) and pzcor > -4 and pzcor <= 4 ] [ sprout 1 [ set shape "box" set color white set size .5 ] ] ask patches with [ pzcor = 4 and pycor < 38 and pycor >= 36 and abs pxcor <= 12 ] [ sprout 1 [ set shape "box" set color white set size .5 ] ] end to setup-turtles crt 1 [ ;;create ball set size .75 set color orange set shape "circle" setxyz 0 0 -3.125 set ball self ] foreach n-values num-takers [ ? * 2 ] [ ;;create num-takers penalty takers create-takers 1 [ set size 6 set color red set shape "person" setxyz 36 (35 - ?) -1 set heading 0 ] ] foreach n-values (num-takers + 1) [ ? * 2 ] [ ;;create num-takers + 1 goalkeepers create-keepers 1 [ set size 6 set color yellow set shape "person" setxyz -36 (35 - ?) -1 set heading 0 ] ] if not Manual [ ask (turtle-set keepers takers) [ set choices n-values (memory-size * 2) [ one-of (list -1 1) ] ;;randomize previous choices set results n-values (memory-size * 2) [ one-of (list 0 0 1 1 1 1 1 1 1 1) ] ;;randomize previous results to success rate of 80% learning-setup ;;setup strategies ] ] get-ready end to learning-setup set strategies n-values num-strategies [random-strategy] set best-strategy first strategies end to get-ready ask (turtle-set keepers takers) [ fd 2 ] ;;dequeue ask keepers with [ patch-here = patch -36 37 -1 ] [ ;;(head of the keepers queue) set gk self move-to patch 0 36 -1 ] ask takers with [ patch-here = patch 36 37 -1 ] [ ;;(head of the takers queue) set tk self move-to patch -6 -10 -1 ] if not Manual [ ask (turtle-set gk tk) [ scout ] ;;obtain scouting data ] ask gk [ save-decision ] ;;both players decide before shot ask tk [ shot-decision ] end ;STRATEGY FUNCTIONS ; (Adapted from El Farol Bar) to-report random-strategy report n-values (memory-size + 1) [ 1.0 - random-float 2.0 ] end to-report predict [strategy subhistory] report first strategy + sum (map [?1 * ?2] butfirst strategy subhistory) end to update-strategies [ sheet1 sheet2 ] let best-score (memory-size ^ 3) + 1 foreach strategies [ let score 0 let kick 1 repeat memory-size [ let pred predict ? sublist sheet1 kick (kick + memory-size) set choice pred / (abs pred) set score score + abs (item (kick - 1) sheet2 - choice) set kick kick + 1 ] if (score <= best-score) [ set best-score score set best-strategy ? ] ] end ;DECISION FUNCTIONS to scout ifelse breed = keepers [ set cheat-sheet1 translate [ choices ] of tk [ results ] of tk set cheat-sheet2 [ choices ] of tk ] [ set cheat-sheet1 translate [ choices ] of gk [ results ] of gk set cheat-sheet2 [ choices ] of gk ] end to-report translate [ l1 l2 ] report ( map [ ?1 * ?2 ] l1 l2 ) end to save-decision ifelse not Manual [ update-strategies cheat-sheet1 cheat-sheet2 let raw-choice (predict best-strategy sublist cheat-sheet1 0 memory-size) + .000001 ;;to avoid division by zero ifelse students = "Goalkeepers" [ set choice (raw-choice / abs raw-choice) ] [ ifelse random 100 < teach-nat-tendency [ set choice 1 ] [ set choice -1 ] ] ;;positive = 1 = natural, negative = -1 = non-natural set choices fput choice butlast choices ;;choices are limited to memory-size * 2 set gk-allchoices fput choice gk-allchoices ;;allchoices are unlimited set dive (random 20) * .0005 ;;how far he stretches to get the ball is random, since direction of ball is random except left or right ifelse choice = 1 ;;1 is natural [ set heading -90 ] [ set heading 90 ] ] [ set dive (random 20) * .0005 ifelse random 100 < gk-tendency [ set heading -90 set choice 1 ] [ set heading 90 set choice -1 ] ] end to shot-decision ifelse not Manual [ let raw-choice (sum cheat-sheet1) + one-of (list .000001 (-1 * .000001)) ifelse students = "Penalty Takers" [ set choice (raw-choice / abs raw-choice) ] [ ifelse random 100 < teach-nat-tendency [ set choice 1 ] [ set choice -1 ] ] set choices fput choice butlast choices set tk-allchoices fput choice tk-allchoices ask ball [ ifelse [ choice ] of myself = 1 [ set heading random-float -1 * (atan 12.2 36) set pitch random-float (atan 8.1 36) set power .04 ] [ set heading random-float (atan 12.55 36) set pitch random-float (atan 8.3 36) set power .03 ] ] ] [ ask ball [ ifelse random 100 < tk-tendency [ set heading random-float -1 * (atan 12.2 36) set pitch random-float (atan 8.1 36) set power natural-power ask myself [ set choice 1 ] ] [ set heading random-float (atan 12.55 36) set pitch random-float (atan 8.3 36) set power non-natural-power ask myself [ set choice -1 ] ] ] ] end ;GO FUNCTIONS to go if over? [ ; if ticks mod (num-takers + 1) = 0 [ ; ask (turtle-set keepers takers) [ ; let Z ind-sequence-Z-score choices count-runs choices ; if (abs Z > 2) [ ; set rejections rejections + 1 ; ] ; ] ; ] if not Manual [ ask (turtle-set gk tk) [ ;;update tendencies set tendency length filter [ ? = 1 ] sublist choices 0 memory-size / memory-size ifelse breed = keepers [ set gk-tendencies fput tendency gk-tendencies ] [ set tk-tendencies fput tendency tk-tendencies ] ] ] move-queue get-ready ask ball [ setxyz 0 0 -3.125 ] set over? false if not Manual [ tick ] ] ;;if not over, proceed: ask ball [ fd power ] ask gk [ if [ distance patch 0 0 -10 ] of ball >= 26 and choice * [ choice ] of tk = 1 [ face ball set dive .01 ] ] ;;the keeper stretches for the ball if he is near enough ask gk [ fd dive ] save? score? end to move-queue ask min-one-of keepers [ distance patch 12 36 -4 ] [ move-to patch -36 (35 - num-takers * 2) -1 set heading 0 set pitch 0 ] ask min-one-of takers [ distance patch 0 0 0 ] [ move-to patch 36 (37 - num-takers * 2) -1 set heading 0 ] end to save? ask gk [ if distance ball <= 2 and (abs ([ heading ] of ball - 180)) > ((atan -12 36) - 180) and [ pitch ] of ball < (atan 8 36) [ ;;if the ball is between the posts and the distance between it and the keeper is less than or equal to 2 set misses misses + 1 set saves saves + 1 set over? true set allresults fput 0 allresults if not Manual [ set results fput 0 butlast results ask tk [ set results fput 0 butlast results ] ] ask ball [ set heading 270 - heading fd 1 ;;make sure the ball is parried away ] ] ] end to score? ask ball [ if [ pycor ] of patch-here > 36 and (abs (heading - 180) > (atan -12 36) - 180) and pitch < (atan 8 36) [ ;;if the ball passes the goal-line between the posts set goals goals + 1 set over? true if not Manual [ ask (turtle-set gk tk) [ set results fput 1 butlast results ] ] set allresults fput 1 allresults ] ] ask ball [ if [ pycor ] of patch-here > 36 and ((abs (heading - 180) <= (atan -12 36) - 180) or pitch >= (atan 8 36)) [ ;;air-ball set misses misses + 1 set over? true if not Manual [ ask (turtle-set gk tk) [ set results fput 0 butlast results ] ] set allresults fput 0 allresults ] ] end ;;Non-functioning probability functions ;to-report ind-sequence-Z-score [ s r ] ;;reports Z-score of a sequence s with r runs ; let tot length s ; let lc 0 ; foreach s [ if ? = -1 [ set lc lc + 1 ] ] ; let rc tot - lc ; report (r - (2 * lc * rc / tot) - 1) / sqrt ( (2 * lc * rc * (2 * lc * rc - tot) ) / ((tot ^ 2) * (tot - 1))) ;end ;to-report count-runs [ s ] ; let runs 0 ; let run? false ; let prev first s ; let cnt (length s) - 1 ; foreach (but-first s) [ set cnt cnt - 1 ; if (run? = false and ? = prev) [ set run? true ] ; if (run? = true and (? != prev or cnt = 0)) [ set run? false set runs runs + 1 ] ; set prev ? ] ; report runs ;end

There are 5 versions of this model.

## Attached files

File | Type | Description | Last updated | |
---|---|---|---|---|

Hong_Matt_Slam.pptm | powerpoint | Poster Slam | about 11 years ago, by Matt Hong | Download |

Penalty Kicks in Soccer.png | preview | Matt Le Tissier is the most accomplished penalty taker in soccer. | about 11 years ago, by Matt Hong | Download |

Penaltykicks-FINAL.docx | word | Final Report | about 11 years ago, by Matt Hong | Download |

penaltykicks_Jun2.docx | word | Progress Report 4 | about 11 years ago, by Matt Hong | Download |

PenaltyKicks_May12.doc | word | Progress Report 1 | about 11 years ago, by Matt Hong | Download |

PenaltyKicks_May20.doc.docx | word | Progress Report 2 | about 11 years ago, by Matt Hong | Download |

Penaltykicks_May28.docx | word | Progress Report 3 | about 11 years ago, by Matt Hong | Download |

PenaltyKicksModelProposal.docx | word | Proposal | about 11 years ago, by Matt Hong | Download |

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