Tournament

Advice for foreigners on how Britons walk

“The British have little sense of pavement etiquette, preferring a slalom approach to pedestrian progress. When two strangers approach each other, it often results in the performance of a little gavotte as they double-guess in which direction the other will turn.”

Advice for foreigners on how Britons walk

“Telling people how to walk is simply not British.”

“But on the street? No, we don’t walk on the left or the right. We are British and wander where we will...”

import random

def walk(number_of_walks):
    """
    Simulate people walking along the pavement
    """
    reds = [random.choice('LR') for k in range(number_of_walks)]
    blues = [random.choice('LR') for k in range(number_of_walks)]
    bumps = sum([reds[k] != blues[k] for k in range(number_of_walks)])
    return bumps
					

# Let us plot a large number of these interactions

num_of_walks = 500
num_of_reps = 2000
bins = 20
plt.hist([walk(num_of_walks) for k in range(num_of_reps)], bins=bins);

					

# What if everyone walks on the left?

def walk(number_of_walks):
    """
    Simulate people walking along the pavement
    """
    reds = [random.choice('L') for k in range(number_of_walks)]
    blues = [random.choice('L') for k in range(number_of_walks)]
    bumps = sum([reds[k] != blues[k] for k in range(number_of_walks)])
    return bumps

plt.hist([walk(num_of_walks) for k in range(num_of_reps)], bins=bins);
					

# What if 1 person walks on the left and the other on the right?

def walk(number_of_walks):
    """
    Simulate people walking along the pavement
    """
    reds = [random.choice('L') for k in range(number_of_walks)]
    blues = [random.choice('R') for k in range(number_of_walks)]
    bumps = sum([reds[k] != blues[k] for k in range(number_of_walks)])
    return bumps

plt.hist([walk(num_of_walks) for k in range(num_of_reps)], bins=bins);
					

# Evolutionary dynamics with a mutation rate:
size_of_population = 100  # Number of people
number_of_rounds = 500  # How many rounds
mutation_rate = .05  # Chance of changing strategy
death_rate = .05  # Chance of removal

reds = ['L' for k in range(size_of_population)]
blues = ['L' for k in range(size_of_population)]
red_data = [sum([k == 'L' for k in reds])]
blue_data = [sum([k == 'L' for k in reds])]

for rnd in range(number_of_rounds):  # Loop through rounds
    for j, pair in enumerate(zip(reds, blues)):  # Loop through players

        if random.random() < mutation_rate:  # Check if random change
            reds[j], blues[j] = random.choice('LR'), random.choice('LR')

        if pair[0] != pair[1]:  # If bump
            if random.random() < death_rate:  # If mind change
                reds[j], blues[j] =  blues[j], reds[j]

    red_data.append(sum([k == 'L' for k in reds]))  # Data collection
    blue_data.append(sum([k == 'L' for k in blues]))

plt.plot(red_data);
plt.plot(blue_data);
					

Mathematics can help understand how behaviour emerges.

Golden Balls

TV show from Feb 2008 to Feb 2009 where a jackpot is to be shared between 2 contestants who secretly choose to "split" or "steal".

1. If both players "split", the jackpot is split equally between them.
2. If 1 player "splits" and the other "steals", the stealer gets the whole jackpot and the splitter leaves with nothing.
3. If both players "steal", they both leave with nothing.

Axelrod's computer tournament

Cooperator

Always cooperate.

Defector

Always defect.

Alternator

Start by cooperating and then switch.

Random

Play randomly.

Tit For Tat

Start by cooperating and then do whatever your opponent does in the previous round.

Demonstration

Applications of Game Theory

• Economics
• Biology
• Computer Science

(1928-2015)

Brouwer's fixed point theorem

Operational Research

www.learnaboutor.co.uk

+VincentKnight
@drvinceknight
vknight.org

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