1. Sex Ratios
Illustrates both the power of game theory. As well as how to “test’ game theory model. Albeit not our domain.

2. The sex ratio is the ratio of males to females in a population
Topic we don’t care about -> State definition -> Not a social behavior -> Not even really about humans -> why study? -> analysis is so clean and so powerful that it will serve as a gold standard against which we will judge our future analyses this semester, and will help to illustrate our methods -> IOW, proof of concept

3. While not human social behavior…
We will use sex ratios to illustrate…
How evolution leads to Nash
How Nash can be used to explain a puzzle
How one can show evidence for a game theory explanation

4. Sex Ratios Are Approximately 50:50 In Most Species
The descent of man (1871)
Don't highlight exception

5. Including Humans

6. Why are sex ratios at birth approximately 50:50 in most species?
Puzzle:
Why are sex ratios at birth approximately 50:50 in most species?
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Let's start by examining some possible explanations that we've heard or read about.

7. Is it to ensure that everyone has a mate?
This one was the one given to me by my 9th grade biology teacher. You might have heard it as well.

8. Makes sense until -> in many species males die off faster -> riskier foraging behavior or fight more -> so that by maturity, few males
From: Trivers 1976

9. Is it to maximize the size of the species?

10. Sex ratio at birth is 51%
No.

11. The real answer, thanks to Fisher (1930)

12. Simplest case:
• Female and male babies require approximately the same parental investment
• No inbreeding

13. FACT: Every child must have exactly one mother and one father

14. Suppose there are: 100 parents: 75 females, and 25 males 100 offspring Each male expects 100/25 = 4 offspring Each female expects 100/75 = 1.33 offspring Males have more offspring
Start w/ “let’s look at the implication of this fact.”
These 75 females have 100 offspring. These 25 males have these same 100 offspring.

15. Parents of males have more grandkids
Holding the number of kids you have constant

16. If everyone else gives birth to offspring at a ratio of 1 male to every 3 females (25:75) You’d do better by having more male offspring 25:75 is not a Nash Equilibrium
Now let's look at some arbitrary sex ratio that's not 50:50. If everyone else is having kids at this ratio. Should you?
Recall the definition of Nash. It says that you're in NE if no one wants to deviate. You did, so we must not be in Nash

17. Same argument holds for any sex ratio

18. Except 50:50 Then, male and female offspring have the same expected number of offspring And you can’t do better by having more male or female offspring
Think back to slide where we had 100 offspring. Now we have 50 dads and 50 moms, and… each has 2 kids.

19. 50:50 is the unique Nash Equilibrium

20. Are we sure evolution will lead to 50:50?

21. If everyone gives birth to offspring at a ratio of 3 females for every male (25:75) A mutant gene arises that leads to more male offspring The individual with the mutant gene will have more grandkids, and the gene will end up disproportionally represented in two generations Since individuals with this gene have more male offspring, this would increase the sex ratio

22. Until… the population hits 50:50 At 50:50, the mutant’s male offspring won’t give more grandkids And the mutant will stop spreading

23. At 50:50, no mutant does better than existing population And if it does by chance, the mutant will do worse and we’ll return to 50:50

24. Evolution leads to Nash Equilibrium

25. Robust
Doesn’t matter if: Males die before maturity, since result is driven by the expected number of children Polygyny, for same reason Population size isn’t constant Sex determined by one sex

26. Wait a minute. Sex ratios aren’t always 50:50!

27. Sex Ratios vs. Parental Investment
Source: Trivers 1976
Explain in detail.

28. Sex Ratios Amongst Inbred Wasps
Source: Herre 1987

29. These exceptions are actually predicted by the theory
Recall the assumptions we made at the beginning. Let's break them one at a time

30. What if parents invest more in one sex?

31. Suppose females twice as costly to make as males

32. Is 50:50 an equilibrium?

33. Male and female offspring yield the same number of grandkids But male offspring cost ½ as much
Intuitively, you can already see that 50:50 won't be an equilibrium, but let's work through the NE logic again

34. If everyone else gives birth to offspring at a ratio of one male to every female (50:50) Can do better by having primarily male offspring, and thus more offspring 50:50 is not a Nash Equilibrium

35. In equilibrium, would have more males

36. Novel prediction that fits the data!

37. What if there is inbreeding?

38. Male offspring mate only with their sisters
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Ask class for answer, then repeat it when right

39. Maximize number of grandkids by having mostly females
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Just need one male offspring

40. Novel prediction that fits the data!

41. Our analysis of sex ratios is the gold standard for treating game theory as a science
As said at beginning, incredible example of how GT can help explain world around us, and help illustrates our methods. Let's go back and summarize what we did.

42. Why are sex ratios at birth approximately 50:50 in most species?
Identify A Puzzle:
Why are sex ratios at birth approximately 50:50 in most species?

43. Use Data to Eliminate Alternative Explanations
Ensure equal ratio at maturity?
Maximize size of species?
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Looked at alternatives

44. Use Data to Eliminate Alternative Explanations
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And eliminated them by examining data from the world around us

45. Use game theory to solve the puzzle (In a way that is depends on reasonable assumptions, can be verified, and is robust!)

46. Use the theory to find novel predictions (not predicted by alternatives)
If females require more parental investment, there will be more males in equilibrium

47. Find Empirical Evidence to Support It

48. Next class… Learning