Question

Suppose 4 people each start with $10. In each of 2 periods, each person flips a coin (50% probability of heads or tails).

a. Suppose that each time a person flips heads, their wealth increases by 70%. If they flip tails, their wealth decreases by 30%. On average, what fraction (or percentage) of the total wealth would you expect the richest person to have in each period? (Show how you calculated the answer.) (1 pt)

b. Now, repeat the exercise in a. above, but with tails decreasing wealth by 70% (heads still increasing wealth by 70%). On average, what fraction (or percentage) of the total wealth would you expect the richest person to have in each period? (Show how you calculated the answer.) (0.5 pts)

c. In class, we looked at the same game, but with heads increasing wealth by 50% and tails reducing it by 50%. In that exercise, we found that the richest person had 37.5% of the wealth at the end of period 1, and 56.3% of the wealth at the end of second period. Which of the two versions of the game you played (a. or b.) resulted in a more similar pattern to this? (0.5 pts)

d. Bonus: (0.5 pts) What explains (2 sentences max.) your answer to c. above? (i.e. what is the key difference or similarity between the games in a. and b., relative to the one in class?)

Answer 1

a)

Person	Remarks	Initial $	Flip Combination	Period 1 ($)	Period 2 ($)
1	Richest ( Luckiest)	10	Heads - Heads	17	28.9
2		10	Heads- Tails	17	11.9
3		10	Tails - Heads	7	11.9
4	Poorest (Unluckiest)	10	Tails - Tails	7	4.9
		Total Wealth	48	57.6
		Richest Person %	35.42	50.17

b)

Person	Remarks	Initial $	Flip Combination	Period 1 ($)	Period 2 ($)
1	Richest ( Luckiest)	10	HH	17	28.9
2		10	HT	17	5.1
3		10	TH	3	5.1
4	Poorest (Unluckiest)	10	TT	3	0.9
		Total Wealth	40	40
		Richest Person %	42.5	72.25

c)

Person	Remarks	Initial $	Flip Combination	Period 1	Period 2
1	Richest ( Luckiest)	10	Heads - Heads	15	22.5
2		10	Heads- Tails	15	7.5
3		10	Tails - Heads	5	7.5
4	Poorest (Unluckiest)	10	Tails - Tails	5	2.5
		Total Wealth	40	40
		Richest Person %	37.5	56.25

The game played in (b) is more similar to the one played in (c) as the total wealth in each period remains the same in both games ($40 + $ 40 = $80), as the wealth addition and deletion is same in both games (70% - 70% in b) and (50% - 50% in c).

(d) The key difference between games played in (a) and (b) is the outcome for flipping heads and tails in (a) i.e. 70% addition for heads and 30% diminuition for tails. But the outcome is in same proportion in games (b) & (c).