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## Applied Game Theory Lecture 4

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**Applied Game TheoryLecture 4**Pietro Michiardi**Recap from last lecture**• Last time we formally discussed about mixed strategies and mixed strategy NE • The big idea was that if a player is playing a mixed strategy in equilibrium, then every pure strategy in the mix must also be a best response to what the other side is doing**Tennis Game (recap)**• We’re going to look at a game in the game: assume two players (Venus and Serena), where Serena is at the net VENUS SERENA RIGHT right Viewpoint LEFT left**Tennis Game (recap)**Serena r l • We identified the mixed strategy NE for this game Venus Serena • [(0.7, 0.3) , (0.6, 0.4)] L R l r p* (1-p*) q* (1-q*) p L Venus (1-p) R (1-q) q**Tennis Game (recap)**• How do we actually check that this is indeed an equilibrium? • Let’s verify that in fact p* is BR(q*) • Venus’ payoffs: • Pure strategy L 50*0.6 + 80*0.4 = 0.62 • Pure strategy R 90*0.6 + 20*0.4 = 0.62 • Mix p* 0.7*0.62 + 0.3*0.62 = 0.62 • Venus has no strictly profitablepure-strategy deviation**Tennis Game (recap)**• But is this enough? There are no pure-strategy deviations, but could there be any other mixes? • Any mixed strategy yields a payoff that is a weighted average of the pure strategy payoffs • This already tells us: if you didn’t find any pure-strategy deviations then you’ll not find any other mixes that will be profitable To check if a mixed strategy is a NE we only have to check if there are any pure-strategy profitable deviations**Discussion (1)**• Can anybody suggest some other places where we see randomization or at least mixed strategy equilibria in sporting events? • Pick your own!! • In general, if you listen to sport comments, you’ll be surprised to hear all kind of stories around statistics and tactics • Especially arguing that since there’s a statistically equal chance when randomizing, there must be no point in playing those strategies This is misleading and wrong: why?**Discussion (2)**• Since we’re in a mixed strategy equilibrium, it must be the case that the payoffs are equal • Indeed, if it was not the case, then you shouldn’t be randomizing!!**Discussion (3)**• After the security problems at U.S. and worldwide airports due to high risks of attacks, the need for devices capable of inspecting luggage has raised considerably • The problem is that there are not enough of such machines • Wrong statements have been promoted by local governments: • If we put a check device in NY then all attacks will be shifted to Boston, but if we put a check device in Boston, the attacks will be shifted to yet another city • The claim was that whatever the security countermeasure, it would only shift the problem**Discussion (4)**• The problem with that line of reasoning was that the concept of mixed strategy was not adopted • What if you wouldn’t notify where you would actually put the check devices, which boils down to randomizing? The hard thing to do in practice is how to mimic randomization!!**Dating and income tax declaration**Interpretations to mixed strategies**The Battle of the Sexes (revisited)**Player 2 b a • We already know a lot about this game • There are two pure-strategy NE: (A,a) and (B,b) • We know that there is a problem of coordination • We know that without communication, it is possible (and quite probable) that the two players might fail to coordinate A p Player 1 B (1-p) 1-q q**The Battle of the Sexes (2)**• Let’s find the mixed strategy NE • Any volunteer?**The Battle of the Sexes (3)**• Player 1 perspective, find NE q: • Player 2 perspective, find NE p:**The Battle of the Sexes (3)**• Let’s check that p=2/3 is indeed a BR for Player1:**The Battle of the Sexes (4)**• We just found out that there is no strictly profitable pure-strategy deviation There is no strictly profitable mixed-strategy deviation • The mixed strategy NE is: Player 1 Player 2 1-p q 1-q p**The Battle of the Sexes (5)**• What are the payoffs to players when they play such a mixed strategy NE? • Why are the payoffs so low? • What is the probability for the two players not to meet? Prob(meet) = 2/3*1/3+1/3*2/3=4/9 1- Prob(meet) = 5/9 !!! Player 1 Player 2**The Battle of the Sexes (6)**• This results seems to confirm our intuition that “magically” achieving the pure-strategy NE would be not always possible • So the real question is: why are those players randomizing in such a way that it is not profitable?**Mixed Strategies: Interpretation #2**• Rather than thinking of players actually randomizing over their strategies, we can think of them holding beliefs of what the other players would play • What we’ve done so far is to find those beliefs that make players “indifferent” over what they play since they’re going to obtain the same payoffs**Mixed Strategies:Interpretation #3**• We could actually think in terms of fraction of a population when we discussed mixed strategies • Let’s motivate this line of thinking through an example/game**The Income Tax Game (1)**Tax payer Cheat Honest • Let’s focus on a simultaneous move game (despite in this case it’s not realistic) • The auditor can decide to audit or not a tax payer • The tax payer can decide to be honest or to cheat in declaring income tax • Take a look at the payoffs… A p Auditor N (1-p) 1-q q**The Income Tax Game (2)**Tax payer Cheat Honest • Is there any pure-strategy NE? • Let’s find what is the mixed-strategy NE… • Despite the mathematics exercise looks and is the same as we saw so far, we’ll give it a different interpretation A p Auditor N (1-p) 1-q q**The Income Tax Game (3)**• Mixed strategies NE: Look at tax payers payoffs To find auditors mixing**The Income Tax Game (4)**• From the auditor’s point of view, he/she is going to audit a single tax payer 2/7 of the time • This is actually a randomization (which, by the way is applied by law) • From the tax payer perspective, he/she is going to be honest 2/3 of the time This in reality implies that 2/3rd of the proportion of population is going to pay taxes honestly**The Income Tax Game (5)**• We have been considering so far a randomization of a single player • Instead, now we say that this is a mixture in the population • Mixed strategies can be thought of as not players mixing their pure strategies but as a mix in a large population of which some people are doing one thing and the other group are doing the other**The Income Tax Game (6)**• What could ever be done if one policy maker (e.g. the government) would like to increase the proportion of honest tax payers? • One idea could be for example to “prevent” fraud by increasing the number of years a tax payer would spend in jail if found guilty**The Income Tax Game (7)**Tax payer Cheat Honest • So we changed the payoff matrix • What happens to q*? • What is now the mixed-strategy NE? A p Auditor N (1-p) 1-q q**The Income Tax Game (8)**• Mixed strategies NE:**The Income Tax Game (9)**• What happened? • It looks like the proportion of honest tax payers didn’t change! • NOTE: what determines the equilibrium mix for the column player is the row player’s payoffs!! • What happened to the probability of checking with an audit a single tax payer? • This is good news, as audits cost money to society and having less frequent audits is beneficial for all!!**The Income Tax Game (10)**• What can we actually do to increase the number of honest tax payers? • We could modify the payoffs to auditors • Make audits cheaper • Make more profitable an audit • We could abandon the idea of Game Theory and just set the probability of audits “out of band” • What would be the problem here?**To sum up**• Lesson 1: mixed strategies can have different interpretations (fraction of population) • Lesson 2: we can verify a mixed strategy NE is effectively one simply by checking pure-strategy deviations • Lesson 3: Row players’ payoffs impact Column players’ mixing probability and vice-versa**Building up on the last interpretations of mixed strategies**From Game theory to evolution**Evolution (1)**• Concept related to a specific branch of Biology • Relates to the evolution of the spices in nature • Powerful modeling tool that has received a lot of attention lately by the computer science community • Why look at evolution in the context of Game Theory?**Evolution (2)**• Game Theory had a tremendous influence on evolutionary Biology • Study animal behavior and use GT to understand population dynamics • Idea: • Relate strategies to phenotypes of genes • Relate payoffs to genetic fitness • Strategies that do well “grow”, those that obtain lower payoffs “die out” • Important note: • Strategies are hardwired, they are not chosen by players**Evolution (3)**• Examples: • Group of lions deciding whether to attack in group an antelope • Ants deciding to respond to an attack of a spider • Mobile ad hoc networks • TCP variations • P2P applications**Evolution (4)**• Evolutionary biology had a great influence on Game Theory • Similar ideas as before, relate strategies and payoffs to genes and fitness • Example: • Firms in a competitive market • Firms are bounded, they can’t compute the best response, but have rules of thumbs and adopt hardwired (consistent) strategies • Survival of the fittest == rise of firms with low costs and high profits**Simplifying assumptions**• When studying evolution through the lenses of GT, we need to make some assumptions to make our life easy • We will relax these assumptions later on • Within spices competition • We assume no mixture of population: ants with ants, lions with lions • Asexual reproduction • We assume no gene redistribution**Evolutionary Game Theory (1)A simple model**• We will look at simple games at first • Two player symmetric games: all players have the same strategies and the same payoff structure • We will assume random tournaments • In a large population of individuals, we pick two individuals at random and we make them play the symmetric game • The player adopting the strategy yielding higher payoff will survive (and eventually gain new elements) whereas the player who “lost” the game will “die out”**Evolutionary Game Theory (2)A simple model**• Assume a large population of players with hardwired strategies • We suppose the entire population play strategy s • We then assume a mutation happens, and a small group of individuals start playing strategy s’ • The question we will ask is whether the mutants will survive and grow or if they will eventually die out**Evolutionary Game Theory (3)A simple model**• StudyEvolutionarily Stable (ES) strategies • Notes: • With our assumptions we start with a large fraction of players adopting strategy s and a small portion using strategy s’ • In random matching, the probability for a player using s to meet another player using s is high, whereas meeting a player using s’ is low**A simple example (1)**Player 2 Defect Cooperate • Have you already seen this game? • Examples: • Lions hunting in a cooperative group • Ants defending the nest in a cooperative group • Question: is cooperation evolutionarily stable? C Player 1 D 1- ε ε**A simple example (2)**Player strategyhardwired C “Spatial Game” All players are cooperativeand get a payoff of 2 What happens with amutation?**A simple example (3)**Player strategyhardwired C Player strategyhardwired D • Focus your attention on thisrandom “tournament”: • Cooperating player will obtaina payoff of 0 • Defecting player will obtain apayoff of 3Survival of the fittest:D wins over C**A simple example (4)**Player strategyhardwired C Player strategyhardwired D**A simple example (5)**Player strategyhardwired C Player strategyhardwired D**A simple example (6)**Player strategyhardwired C Player strategyhardwired D A small initial mutation israpidly expanding instead ofdying outLet’s now try to be a little bitmore formal**A simple example (7)**Player 2 Defect Cooperate • Is cooperation ES? C vs. [(1-ε)C + εD] (1-ε)2 + ε0 = 2(1-ε) D vs. [(1-ε)C + εD] (1-ε)3 + ε1 = 3(1-ε)+ ε 3(1-ε)+ ε > 2(1-ε) C is not ES because the average payoff to C is lower than the average payoff to D C Player 1 D ε For C being a majority 1- ε ε 1- ε For D being a majority**A simple example (8)**Player 2 Defect Cooperate • Is defection ES? D vs. [εC + (1-ε)D] (1-ε)1 + ε3 = (1-ε)+3ε C vs. [εC + (1-ε)D] (1-ε)0 + ε2 = 2ε (1-ε)+3 > 2 ε D is ES: any mutation from D gets wiped out! C Player 1 D ε For C being a majority 1- ε ε 1- ε For D being a majority**Observations**• Lesson 1: Nature can suck • It looks like animals don’t cooperate, but we’ve seen so many documentaries showing the opposite!!! Why? • Sexual reproduction, and gene redistribution might help here • Lesson 2: If a strategy is strictly dominated then it is not Evolutionarily Stable • The strictly dominant strategy will be a successful mutation**Another example (1)**a b c • 2 player symmetric game with 3 strategies • Is “c” ES? c vs. [(1-ε)c + εb] (1-ε) 0 + ε 1 = ε b vs. [(1-ε)c + εb] (1-ε) 1 + ε 0 = 1- ε 1- ε > ε “c” is not evolutionary stable, as “b” can invade it a b c

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