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Game Theory Basic Concept, Game Theory for Blockchain

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- 1. ‘Economic Design’ in Cryptoeconomics II. Game Theory Basics Jongseung Kim(deframing@gmail.com) Director, SK telecom Blockchain Business Development Unit 2018. 9. 27
- 2. Outline • Strategic Form / Extensive Form Games • Dominated / Dominant Strategies • Nash Equilibrium / Focal Point • Nash Equilibrium for Blockchain • Blockchain and Coordination Games 2
- 3. Strategic Form Games1 3 • A strategic form game (also called a normal form game) is composed of three elements: - a set of individuals, called players, denote N = {i1, i2, …, in}; - for each player i ∈ N, a pure strategy, denoted Si; and - for each player i ∈ N, a payoff function ui, that gives the payoff player i will receive for all possible combinations of actions chosen by all the players. So ui is a function from S = S1 × S2 × Sn to R. • A game is denoted G = <N, (Si)i∈N, (ui)i∈N>. A strategy profile s = (s1, …, sn), also called an outcome, is a collection of strategies (one for each player). When the action set of every player is finite, then we say that the game is finite. • A Strategy that consists of a randomization over pure strategies is called a mixed strategy. - A mixed strategy for a player i is a probability distribution over her set of pure strategies. 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018
- 4. Extensive Form Games1 4 • Extensive form games allow capture of situations where the players may observe the play of other players before they have to play. The key concept is that of history, which simply captures the sequence of actions taken by the players. • A extensive form game of perfect information is composed of the following elements: - a set of players, denote N = {i1, i2, …, in}. - a set of histories, denoted H. A typical history is a sequence of actions taken by the players. - a function P that assigns to each nonterminal history h a player in N. - for each player i ∈ N, a payoff function ui, that gives the payoff player i will receive at each terminal history. • A strategy of a player i in an extensive form game is a function that assigns, for each nonterminal history h such that P(h) = i, an action in A(h). 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018
- 5. Imperfect Information1 5 • Extensive form games can accommodate broader situations, where players may only have an imperfect knowledge of the history of play. • A extensive form game of perfect information is composed of the following elements: - a set of players, denote N. - a set of histories. - a function P that assigns a player to each nonterminal history. - for each player i ∈ N, a partition of all the histories h such that P(h) = i. If two histories h and h’ are in the same element of the partition, then A(h) = A(h’). An element of the partition is called an information set. - for each player, a payoff function that gives the payoff player will receive at each terminal history. 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018
- 6. Dominated Strategies1 6 • A pure strategy si is strictly dominated for player i if there exists a mixed strategy σi for player i such that - ui(σi, s-i) > ui(si, s-i) for all si ∈ S-i. • A pure strategy si is weakly dominated for player i if there exists a mixed strategy σi for player i such that - ui(σi, s-i) ≥ ui(si, s-i) for all si ∈ S-i - and there exists at least one pure strategy profile s-i such that ui(σi, s-i) > ui(si, s-i) 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018
- 7. Dominant Strategies1 7 • A pure strategy si is strictly dominant for player i if, for any mixed strategy σi for player i, - ui(si, s-i) > ui(σi, s-i) for all si ∈ S-i. • A pure strategy si is weakly dominant for player i if, for any a mixed strategy σi for player i, - ui(si, s-i) ≥ ui(σi, s-i) for all si ∈ S-i - and there exists at least one pure strategy profile si such that ui(σi, s-i) > ui(si, s-i). • If a player’s strategy is strictly dominant, then all the other strategies of that player are strictly dominated. • If a player’s strategy is weakly dominant, then all the other strategies of that player are weakly dominated. 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018
- 8. What is Equilibrium?1 8 • An equilibrium is a situation in which some motion or activity or adjustment or response had died away, leaving something stationary, at rest, “in balance”, or in which several things that have been interacting, adjusting to each other and to each other’s adjustment, are at last adjusted, in balance, at rest. • An equilibrium can be exact or approximate. It can be always approached but never quite achieved, the potential equilibrium itself continually changing. And equilibrium can be partial or more complete, short run or long run. • An equilibrium is simply a result. It is what is there after something has settled down, if something ever does settle down. The idea of equilibrium is an acknowledgement that there are adjustment processes. • There may be many things wrong with “equilibrium analysis”, including the possibility that it oversimplifies by neglecting processes of adjustment, or exaggerates the prevalence of equilibrium by neglecting shifts in the parameters that determine the equilibrium. 1Thomas C. Schelling, <Micromotives and Macrobehavior>W. W. Norton & Company, 2006
- 9. The Austrian School Approach to Equilibrium Analysis 9 • An equilibrium is a continually moving target.1 - The market economy is a coordination mechanism that enables individuals to make use of the information they possess to plan their economic activities in such a way that they are consistent with everyone else's plans. • Hayek depicts economic equilibrium as the coordination of individual plans. In the short run this means that markets clear, so that the quantity supplied equals the quantity demanded in all markets.2 • Kirzner depicts equilibrium as the absence of unexploited profit opportunities.3 1Randall G. Holcombe, <Advanced Introductionto the Austrian School of Economics> Edward Elgar Pub, 2014 2FriedrichA. Hayek, <Individualism andEconomic Order> University of Chicago Press, 1996 3Israel Kirzner, <Competitionand Entrepreneurship>LibertyFund Inc., 2013
- 10. Nash Equilibrium 10 • A Nash Equilibrium of a game <N, (Si)i∈N, (ui)i∈N> is a profile of strategies σ such that for each player i ∈ N we have - ui(σi, σ-i) ≥ ui(σ’i, σ-i) for each σ’i ∈ ∆(Si).1 • Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy.2 • Any game with a finite set of players and a finite number of pure strategies always admits a Nash Equilibrium (possibly in mixed strategies). 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018 2https://en.wikipedia.org/wiki/Nash_equilibrium
- 11. Prisoner’s Dilemma 11 1http://studycas.com/component/k2/revisiting-nash-equilibrium-in-prisoner-s-dilemma If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium. Prisoner's Dilemma – Payoff Matrix Prisoner’s Dilemma Payoff Matrix1
- 12. Bayesian Games : Games with Incomplete Information1 12 • In most real-life situations individuals do not know everything about the persons they interact with. Game with incomplete information, also called Bayesian games, are designed to capture such situations. • A Bayesian game consists of: - a finite set of individuals, called players, denote N = {i1, i2, …, in}; - for each player i ∈ N, a set of actions, denoted Ai; - for each player i, a set Ti of possible signals; - a probability distribution f over the set of signal profiles T = T1 × T2 × … × Tn; - for each player i ∈ N, a payoff function ui, that gives the payoff player i will receive for each possible combination of actions chosen by all the players and each possible profile of signals received by the players. So ui is a function from A × T to R, where A = A1 × A2 × … × An. 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018
- 13. Bayesian-Nash Equilibrium 13 • A profile of (pure) strategies s is a Bayesian-Nash Equilibrium if for each player i and each signal ti ∈ Ti, and for each action ai ∈ Ai, - E[ ui((si(ti), s-i(t-i )), (ti, t-i)) | ti ] ≥ E[ ui(ai, s-i(t-i ), (ti, t-i)) | ti ]1 • A Bayesian-Nash equilibrium is defined as a strategy profile and beliefs specified for each player about the types of the other players that maximizes the expected payoff for each player given their beliefs about the other players' types and given the strategies played by the other players.2 • This solution concept yields an abundance of equilibria in dynamic games when no further restrictions are placed on players' beliefs. This makes Bayesian-Nash equilibrium an incomplete tool with which to analyze dynamic games of incomplete information. 1Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018 2https://en.wikipedia.org/wiki/Bayesian_game
- 14. What is Schelling Point? 14 1https://en.wikipedia.org/wiki/Focal_point_(game_theory) Thomas Schelling describes "focal point[s] for each person's expectation of what the other expects him to expect to be expected to do”1
- 15. Nash Equilibrium and Blockchain Platforms1 15 • Nash Equilibrium is an incredibly useful concept for individuals working in blockchain. • While many of those working in the space recognize the importance of game theory, few are demonstrating — in their whitepapers or elsewhere — that the behaviors their platforms are created to induce are indeed equilibria on those platforms. • A common mistake that organizations make when trying to apply game theoretic modeling is that they assume payoffs = tokens. • In order to accurately represent a strategic situation, payoffs must capture (to the extent possible) all the gains that participants get from each action combination — including non-monetary payoffs (such as reputation) and payoffs from outside the platform. 1https://medium.com/prysmeconomics/nash-equilibrium-and-blockchain-d6a6f47a7a37
- 16. Why does Nash Equilibrium matter for Blockchain? 1 16 • Strategic games are everywhere in blockchain ecosystems — in the design of markets, in consensus algorithms, and in the decisions that users and miners make of whether to act in the best interest of the platform community. • Users and miners participating in blockchain ecosystems are not controlled by a centralized authority: they act in the way that maximizes their own outcomes, taking into consideration what they expect the other participants to do, exactly as game theory specifies. • Developers create a game when they decide what actions are available to participants and how these actions will be rewarded in tokens or other outcomes. They need to design their platforms so that the outcomes they want to occur are equilibria of the strategic games that their participants are playing. 1https://medium.com/prysmeconomics/nash-equilibrium-and-blockchain-d6a6f47a7a37
- 17. Token Curated Registries Case1 17 • An example of this type of game design is token curated registries, which aim to use game theory to decentralized the curation of ranked lists. • Rather than relying on a list owner (such as a newspaper editorial team) to rank items by quality, users of the registry vote on which content should be accepted to the list. • The token economics are supposed to guarantee that users only accept high-quality content to the list and don’t accept low-quality content. • The claim is that since users must buy tokens to submit content to the list and stake tokens to vote on content, and the token will increase in value as the list becomes more popular, users will want to maximize the popularity of the list, and they will do this by voting for the highest-quality content. 1https://medium.com/prysmeconomics/nash-equilibrium-and-blockchain-d6a6f47a7a37
- 18. General Lessons for Blockchain Organizations1 18 • Lesson 1: It is not enough to state that an outcome is an equilibrium — you must show that that is the case. • Lesson 2: Before determining the equilibrium, you need to outline what the possible actions are for each participant, and their payoffs from choosing each. • Lesson 3: After outlining the possible actions of the participants and their payoffs from each, the developer must verify that the desired outcome is an equilibrium of the game. • Lesson 4: Many games have multiple equilibria — you need to verify that the equilibrium you want to occur will be selected. 1https://medium.com/prysmeconomics/nash-equilibrium-and-blockchain-d6a6f47a7a37
- 19. Blockchain and Coordination Games1 19 • When there are multiple equilibria, all players in the game need to coordinate on the same Nash equilibrium in order for the benefits of equilibrium to be realized. This is why these games are called coordination games. • Understanding coordination games is important for blockchain developers because they arise frequently and, without proper design, they can lead to potentially undesirable outcomes such as hard forks or non- equilibrium outcomes. • Two types of problems can occur in a coordination game. - A coordination failure takes place when the players of a strategic game do not land on an equilibrium outcome. - A coordination fault is when the equilibria of the game can be ranked — everyone agrees that one is better than another — but the players coordinate on a suboptimal equilibrium. 1 https://medium.com/prysmeconomics/blockchain-and-coordination-games-failures-and-focal-points-e166cc244973
- 20. Why Coordination Games are important for the Blockchain?1 20 • In the context of blockchain, upgrading the system is often a coordination game. • Suppose that a blockchain community is considering a policy upgrade. (Stay, Upgrade, Hard fork) • Which path a user prefers depends on two forces: - Their relative preferences over the two options — how much they care about the policy upgrade. - Their preferences with respect to network effects — how much they care about being on the same chain as other users. • Because my preferred choice depends on what the other users choose to do, this is a coordination game. • A focal point is a public signal that all players understand. A focal point indicates to everyone which Nash equilibrium they should choose. A focal point can be anything. 1 https://medium.com/prysmeconomics/blockchain-and-coordination-games-failures-and-focal-points-e166cc244973
- 21. Governance System for Coordination Games1 21 • In the blockchain upgrade situation, a governance system could help facilitate the coordination of the community on an equilibrium that everyone can accept. • This system would only work if the community believed that other users would abide by the outcome. - the governance system serves as a focal point. - It does not enforce behavior; rather, it guides users in helping them make coordinated choices so that better outcomes can be reached. • Introducing a system of rules for determining whether an upgrade should be implemented by way of a properly structured governance system will reduce the likelihood of a hard fork coordination failures. • When designing a platform, it is important to recognize when a system has multiple equilibria so that measures can be put in place that will help coordinate the actions of users on good equilibria. 1 https://medium.com/prysmeconomics/blockchain-and-coordination-games-failures-and-focal-points-e166cc244973
- 22. References • Guillaume Haeringer, <Market Design : Auction and Matching> The MIT Press, 2018 • Randall G. Holcombe, <Advanced Introduction to the Austrian School of Economics> Edward Elgar Pub, 2014 • Thomas C. Schelling, <Micromotives and Macrobehavior> W. W. Norton & Company, 2006 • Y. Narahari, <Game Theory and Mechanism Design> World Scientific Publishing Company, 2014 • http://studycas.com/component/k2/revisiting-nash-equilibrium-in-prisoner-s-dilemma • https://en.wikipedia.org/wiki/Focal_point_(game_theory) • https://en.wikipedia.org/wiki/Nash_equilibrium • https://en.wikipedia.org/wiki/Bayesian_game • https://medium.com/prysmeconomics/nash-equilibrium-and-blockchain-d6a6f47a7a37 • https://medium.com/prysmeconomics/blockchain-and-coordination-games-failures-and-focal-points-e166cc244973 22