Applied Game Theory Lecture 4. Pietro Michiardi. Recap from last lecture. Last time we formally discussed about mixed strategies and mixed strategy NE
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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