Conference 4Intermediate Microeconomics - Fall 201824/09/2018Problem 1: Real life prisoners’ dilemna.1. Can you think of any prisoners’ dilemna like situations that happen in the real life?Explain.2. We are going to think about two examples of real life prisoners’ dilemna.High Frequency Trading (HFT). Follow the link on Moodle to the explanatory video.What can we talk about an arms’ race and how does this relate to a prisoners’dilemna?Bitcoin rests on Blockchain, a system that claims to have solved the ByzantineGenerals’ problem. Follow the link on Moodle to the video on this topic, and thenread Satoshi Nakamoto’s (the blockchain’s inventor) explanation on how to solve it.How is the Byzantine General’s problem analogous to a prisoners’ dilemna? Howdoes blockchain solve it?High frequency trading (HFT) is just automated computer trading. People in financetalk about HFT’s arms race. Why? Because there are two ways for banks and fundsto get an edge over their competitors in HFT: the first is to design a better algorithm,and the second is to get a faster access to market exchanges. To achieve this, banksand funds invest massively in optic fiber, radio transmitter, etc. to get to and receivefrom exchanges as quickly as possible.Let us consider a game in which there are two banks that each have two actions:invest at a cost in better transmitting equipment (I), or not invest (N). Investinggives an edge over the competitor, but if both bank invest, they loose that edge.Therefore the game they are playing could look like this:I NI (1, 1) (4, 0)N (0, 4) (2, 2)Just like in the prisoners’ dilemna, the equilibrium is (I, I), while both firms wouldhave been better off not spending money to invest, at (N, N ).Bitcoin. The Byzantine’s general problem can be summed up as follows: only anattack carried out simultaneously by all generals can be successful, but generals haveno way of communicating safely to agree on when to attack. Because any messen-ger they send must cross enemy’s territory, they can never be sure that the others1generals received their message. And even if they did receive it, their confirmationmessage might get lost too.This situation looks like a prisoners’ dilemna because generals are likely to decidethat their dominant strategy is to retreat: if they attack alone, they will surely diein the battle.Satoshi Nakamoto sets up the Byzantine problem in a computing framework, butthe analogy with the prisoners’ dilemna remains. Blockchain solves it by allowing forsafe communication. Before attacking, generals must solve a series of problems called“proof-of-work”. Once a proof-of-work problem is solved, it’s solution is included in ablock that is linked to the previous blocks (hence the name “blockchain”). Then theymove on to the next proof-of-work problem, keeping in mind the previous solutions.Because a proof-of-work problem takes on average 10 minutes to solve, after 2 hourseach of the 12 generals should have solved one problem. If this is the case, thengenerals know they all worked on the same chain, and have all seen it, therefore theycan attack simultaneously at the time outputted by the last block.Problem 2: Elimination of dominated strategiesLeft Middle RightUp (4, 3) (2, 7) (0, 4)Down (5, 0) (5, −1) (−4, −2)1. Use successive elimination of diminated strategies to solve the game. Explain the stepsyou followed.The steps are the following:Middle>Right so that we can eliminate Right:Left MiddleUp (4, 3) (2, 7)Down (5, 0) (5, −1)Down>UpLeft MiddleDown (5, 0) (5, −1)Left>Middle. Hence (Left,Down) is a candidate for Nash equilibrium.2. Show that your solution is a Nash equilibriumWe are going to check that Down is a best response to Left and Left a best response toDown (i.e. that no player wants to deviate).If player 2 plays Left, then player 1 has no incentive to deviate from Down to Upsince if she does, she will get a payoff of 4 instead of 5.If player 1 plays Down, then player 2’s next best payoff is -1 if he plays Middle,instead of 0 is he sticks to Left.2Problem 3: Completing payoffsLeft Middle RightUp (4, ) ( , 2) (3, 1)Middle (3, 5) (2, ) (2, 3)Down ( , 3) (3, 4) (4, 2)1. Complete the payoffs of the game table above so that player column has a dominantstragegy. State which stragegy is dominant and explain why. (There are several possibleanswers)Left Middle RightUp (4, 1) (8, 2) (3, 1)Middle (3, 5) (2, 6) (2, 3)Down (2, 3) (3, 4) (4, 2)Then Middle is a dominant strategy for the column player.2. Complete the payoffs of the game above so that neither player has a dominant stragegy,but also so that each player does have a dominated strategy. State which strategy aredominated and explain why. (There are several possible answers)Left Middle RightUp (4, 2) (4, 2) (3, 1)Middle (3, 5) (2, 2) (2, 3)Down (4, 3) (3, 4) (4, 2)Then Right is dominated by Left for the column player, and Middle is dominated byDown for the row player.Problem 4: A location gameEach of n people chooses whether or not to become a political candidate , and if so whichposition to take. There is a continuum of citizens, each of whom has a favorite position;the distribution of favorite positions follows a normal distribution on [0, 1] with mean12andvariance 1. A candidate attracts the votes of those citizens whose favorite postions are closer tohis position than the position of any other candidate; if k candidates choose the same positionthen each receives the fraction1kof the votes that the position attracts. the winner of thecompetition is the candidate who receives the most votes. Each person prefers to be the uniquewinning candidate than to tie for first place, perfers to tie for first place than to stay out of thecompetition, and prefers to stay out of the competition than to enter and lose.1. Formulate this situation as a strategic gameHow many players are there?There are n players.3What is the players’ set of actions?Each player can choose to stay out of the game, or enter and position themselves on[0, 1]. Therefore each player’s set of action is Out ∪ {0, 1}What are players’ preferences regarding the game’s outcomes?Each player prefers an action profile in which he obtains more votes than any otherplayer to one in which he ties for the largest number of votes; he prefers an outcomein which he ties for the first place (regardless of the number of candidates with whomhe ties) to one in which he stays out of the