Question 1: Tipping points (50%) Social norms are ubiquitous in human societies. Sometimes they are sud denly abandoned without warning. just based on rumors that start circulating in the society. The new norm may be better or worse for society. Some mcial scientists naively attribute the change in social norms to the presence of deviants in the population, that is, of individuals who defy the norm at their own cost. Yet. often a social norm is tipped over in favor of a new one. even a detrimental one. although there are no deviants in the population. Besides. it is difcult to tell if someone is a deviant or simply a conformist to a new norm] Consider for instance queuing at a bus stop during rush hour. Upon arrival at a bus stop an individual can either queue so that everyone enters the bus on a first come rst served basis. up to the bus capacity and then possibly waiting for the next bus to arrive; or the individual can jump the line. Of course, society establishes reputational norms that make it costly to jump the line. The situation every bus rider faces is then summarized as follows. If a bus rider faces another, their interaction gives rise to the following payoff matrixl t' = normal Q 1, 1.3. 1.8 J 2' = normal J 1.1 Q 0. 1 where J stands for jumping the line', Q for Equeuingl. and each pair of numbers represents bus ridersT payoffs (the rst for bus rider 1}. So for example. if both bus riders queue. they are able to orderly take the bus. quickly and without wasting time in discussions that can turn violent. |The good behavior clearly is to queue. Jumping the line when the other bus riders queue gives benets to a bus rider because they are able to always take the bus. but the jumper incurs a (physical? psychological? monetary?) cost {scorning. scolding. and m on; or a fine}, resulting in a reduced net payoff. When they both jump the line, bus riders fight. and their payoff is lesser than if they both queued. Finally, queuing when the others jump the line results in a zero payoff. because the bus rider cannot catch the bus (on time). A deviant is someone who dees the sanctions of the social norm. and always jumps the line. If any normal bus rider faces a deviant. their interaction leads to -1 = deviant J i = normal J 1. 1 0, 1 Currently there are no deviants in the population. There are nevertheless immi- grants that just came to town. Immigrants are indistinguishable from the local population. Rumors hit that suggest there are deviants among the immigrants (there are none). Now everyone thinks that is possible, with equal probability Pr(f), that a fraction f of bus riders they face is deviant. Specifically, people talk about this fraction to be any of the numbers 0, 1, 2, ...(n -1), for n a large, but finite even number. Some bus riders (type /) will have received the following information 3..{n - 2 n- while the others (type II) will have received (of(1, 2).. {n - 3 n- 2un - n n No bus rider faces a bus rider of her own type. Note that type II bus riders know that there are no deviants. a) Compute the Nash Equilibria prior to the spreading of rumors. Is (Q.Q) an equilibrium? b) (1) "Type II bus riders know that there are no deviants, and they know that type / bus riders know that there are no deviants." (2) "Type II bus riders know that deviants cannot be more than a 1 fraction of the population, and they know that type / bus riders know this." Which of the two statements, if any, is true? Formally explain. c) Explain term by term what the following expression represents: for 1 ken/2, Eu(Q)ly(- n - 2k )) = 1.8Pr(Quin -2k +1 2k +11 II 72 +1.8 Pr(Q y.( n - 2k 2k 1 II n 2 d) Construct the Bayesian Nash Equilibrium after the rumors have spread (but there are still no deviants around, that is, at f = 0). "You need to have deviants in order to tip over to a new norm. Thus, bus riders will not jump the line even if rumors spread." Is this statement correct? e) What happens to the possible fraction of early consumer when n - too? Is the informational change necessary for a norm change small or large? What if the payoff for (Q.Q) becomes 2.2