Intro to Game Theory

Presents an introduction to solution concepts in game theory. Describes how to find dominant strategies, eliminate dominated strategies, and find Nash equilibria. Describes how to find mixed strategy equilibria, and contains a brief introduction to sequential games.

[@sharzilamh]

Thank you for this excellent presentation. At last I'm very clear on how to play this game theory. You are the best!!!

[@KatherineSilzCarson]

Thank you for watching!

[@mihnea1796]

Great presentation, helped me understand game theory basics better than 2 textbooks and a graduate-level lecture :) Thank you very much!

[@KatherineSilzCarson]

Thank you for watching!

[@BitBard302]

This is educational. If you're into it, I highly advocate for a book with corresponding topics. "Game Theory and the Pursuit of Algorithmic Fairness" by Jack Frostwell

[@alaminhussien6505]

KIndly continue your teaching, every single of your video is well explained and beneficial. Thank you so much.

[@KatherineSilzCarson]

Thank you for watching, Alamin!

[@manarlabidi6448]

you just saved a lost soul, I wish they teach us like that... 6 pointless hours in 20mins!

[@KatherineSilzCarson]

Thank you for watching, Manar!

[@TrangNguyen-mq2dk]

Thank you for your helpful lesson. It helps me so much !

[@KatherineSilzCarson]

Thank you for watching!

[@Abcilswd]

You are the best! This lecture has everything I have to master for my exam. The method of how to solve the problem is excellent! Thank you very much!

[@KatherineSilzCarson]

Thank you! I am glad the video was helpful!

[@Abcilswd]

Katherine Silz-Carson Without preview textbook, after I listened to your lecture twice, I practiced the questions in my textbook and my previous exam paper. I answered every game theory question correctly.( I failed the exam last year). I made mistake by comparing the payoff when I was identifying dominant strategy. In addition, I did not know how to eliminate the weak dominated strategy. Thus it was so difficult. Now everything is clear. You make the game theory easy to understand. It is the greatest wisdom of you. Thank you very much. I believe I will pass the exam for sure.

[@KatherineSilzCarson]

Thank you! Good luck on your exam!

[@zekaekberov433]

Thank you very much. İt was extremely helpful.

[@KatherineSilzCarson]

Thank you for watching, Zeka!

[@preetishramasawmy296]

Wow amazing intro to game theory- clear and concise!

[@KatherineSilzCarson]

Thank you for watching, Preetish!

[@guanghongweng2688]

Thanks for coloring gamer1 and gamer2, this help me a lots

[@KatherineSilzCarson]

Thank you for watching!

[@adiskarizkikhayyumi9696]

Thank you so much. You help me a lot :")

[@KatherineSilzCarson]

Thank you for watching!

[@sankadharmapala4064]

very good explanation, Really helpful..... thanks a lot🤩

[@KatherineSilzCarson]

Thank you for watching!

[@somayaalhadi7920]

Thank you so much ... very helpful

[@KatherineSilzCarson]

Thank you for watching, Somaya!

[@Saurabhyadav-yb2oo]

Wonderful explanation mam, keep it up👍👍👍

[@KatherineSilzCarson]

Thank you for watching!

[@sarahdemonia5591]

Hi! I did not understand how you figured out you needed to make the second row "(1-p)" and how you set up the equation starting at around 15:22. Could you explain this? Thank you, great video!

[@KatherineSilzCarson]

Thanks for your question Sarah. The reason why the second row is (1-p) is that player 1 only has two possible actions, top and bottom, and the probability that they play each action must add up to one. Therefore, if player 1 plays Top with probability p (example p = 0.35), then they will play bottom with probability 1 - p = 0.65, since 0.35 + 0.65 = 1. To set up the equation, calculation Player 2's expected payoff from each of their actions. If player 2 plays Left, then they will get 2 with probability p (if player 1 plays Top), and 0 with probability (1-p), so Player 2's expected payoff is 2p + 0(1-p) = 2p. If player 2 plays Right, then they will get 0 with probability p (if Player 1 plays Top) and 1 with probability (1-p) (if player 1 plays Bottom), so player 2's expected payoff = p x 0 + (1-p) x 1 = 1-p. If player 2 is indifferent between left and right, then that means that their expected payoff from left and their expected payoff from right must be equal, which implies that 2p = (1-p). Hope this helps! Thank you for watching!

[@albertloh462]

At 17.06/19.56 timeline, you said "we have found the mixed strategy equilibrium of the game." So what exactly is the mixed strategy equilibrium of the game? Is it Bottom-Left where Pr = 4/9?

[@KatherineSilzCarson]

The mixed strategy equilibrium is both players playing their pure strategies with the probabilities that we solved for. So the equilibrium consists of both players' probabilities and actions. That is, Player 1 plays Top with probability 1/3, and Bottom with probability 2/3, and Player 2 plays Left with probability 2/3 and Right with probability 1/3. Think of the probabilities as each player's equilibrium beliefs about how likely the other player is to play each of their pure strategies. In equilibrium, these beliefs make a player indifferent between their pure strategies. Hope this explanation helps. Thank you for watching!

[@alexzhang2361]

Thank you for the great explanation. Would you mind telling more about 'Pr' at 17:00? Thank you!

[@KatherineSilzCarson]

@Alex - Pr stands for probability. To get the probability of each outcome (e.g. Top, Left; Bottom, Right; etc) just multiply the probability of the Row player's strategy by the probability of the Column player's strategy. For example, the probability that the game ends up at Top, Left is 1/3 x 2/3 = 2/9. Hope this helps! Thank you for watching!

[@saintarj4552]

pretty solid video

[@KatherineSilzCarson]

Thank you!

[@bhavdeepkaur2201]

I don't understand where the numbers in the matrices are coming from? (1,2,0,0)

[@KatherineSilzCarson]

They are designed to reflect a player's preferences. They are made up, but designed to reflect how a player ranks the possible outcomes of a game. If a player likes outcome A better than outcome B, then their payoff from outcome A will be greater than their payoff from outcome B. Hope this helps!

[@laelafitriah3823]

Hi thank your for the presentation, it helps me alot to understand the principle of game theory. But im still confused on „finding the Nash Equilibria of a game“, starting from minute 12.49. In my understanding, Middle - Left = Bottom - Right. But why Middle - Left is Nash Equilibria, meanwhile Bottom - Right is not Nash Equilibria? Bottom - Right : You said: Given that player 2 is playing Right, player 1 has an incentive to swift to center. So, this is not Nash Equilibria. This situation is exactly the same with Middle - Left: Given that player 1 is playing middle, actually player 2 has an incentive to swift to center. Refer to the explanation of Bottom - Right, to my understanding the Middle - Left is not a Nash Equilibria either. Please correct me on this? Thank you so much for your kind reply. but has difference

[@KatherineSilzCarson]

At (Middle, Left), if Player 2 is playing Left, Player 1 makes 1 whether or not they play Middle or Bottom. Thus, because Player 1 cannot do better by switching to Bottom, Player 1 has no incentive to change. Similarly, if Player 1 is playing Middle, Player 2 either makes the same (1) by switching to Center, or less (0) by switching to Right. Because Player 2 cannot do better by switching, Player 2 has no incentive to change their strategy either. Because neither player has an incentive to change their strategy at (Middle, Left), that outcome is a Nash Equilibrium. However, at (Middle, Center), Player 1 can increase their payoff by switching to Bottom. All it takes is one player having an incentive to change their strategy for the outcome to not be a Nash equilibrium. Because (Middle, Center) fails this test for Player 1, that outcome is not a Nash equilibrium. Hope this helps. Thank you for watching!

[@brothdian]

I searched game theory intro and this pops up? Whaaaat?

[@mdsameedkhan]

Wouldn't it be better if you include this in the Game Theory playlist?

[@KatherineSilzCarson]

It probably should be. I think that the reason it's not is that I made the Intro video (which is kind a bit overview) before I made the more detailed individual videos. Thank you for the suggestion!

[@QUBIQUBED]

​@@KatherineSilzCarsonI don't understand what the sequential diagram means.

[@KatherineSilzCarson]

@@QUBIQUBED The sequential diagram tells you the order of players, the possible actions of players, and the payoffs when one player goes first, then the second player makers their decision, etc. The person at the top or left-hand side of the diagram goes first. The branches coming out from the dot for that player tells you the possible things the player can do. The dots at the end of those branches tells you what players goes next, and the branches tell you what they can do, etc. The bottom of the diagram tells you each player's payoff from that outcome of the game. The first player's payoff is listed first, the second player's payoff is listed second, etc. Hope this helps! Thank you for watching!

[@nattamonsuk]

Thank you for very clear explanation

[@KatherineSilzCarson]

I am glad that you found it helpful. Thank you for watching!

[@nihadrustmsoy6212]

Thank you for helpful lectures which are more useful and short at once than professor's bla bla. But a question: Why does Player 2 has an incentive to change to Center-Bottom from Center-left? 13:44

[@KatherineSilzCarson]

They don't. Player 2 can choose either Left, Center, or Right. If Player 1 chooses Bottom, then Player 2 makes 1 if they choose either left or Center, and 2 if they choose Right. Thus, they will choose right.Hope this helps! Thank you for watching!

[@duhtraders6477]

How big and complex can you make the game tables?

[@KatherineSilzCarson]

Certainly more complex than the ones in this video, but I would say three players is the limit (then you have to have one table for each of the 3rd player's strategies). In principle, you can have many rows and columns.

[@duhtraders6477]

@@KatherineSilzCarson I am new to game theory, I am hoping to draft a game relative to day traders. I must be clear, this is only meant for individuals opening and closing stock positions during a single day and it doesn't pertain to the market or markets as a whole. In a day trade any participant can go long or go short, with 3 types of players, retail traders, institutional traders and the market maker. All 3 players can be forced to both buy and sell but the market maker cannot go short. Is there any thing you know that I can use to model this?

[@KatherineSilzCarson]

@@duhtraders6477 It sounds like you have the basic framework already. One challenge I see is that, as you describe their possible strategies, the traders don't always have a choice about what they do (at least that is how I interpret the word "forced.") The one thing that you need to think about now are how their potential payoffs are related ie. if one trader does X and the other does Y, how does Z's payoff change depending on what Z chooses to do. It may be that rather than using a table, creating a function that describes each player's payoffs as a function of the other players' actions might make more sense.

[@TerrelleCheers1]

"Argument" was found ✓

[@rickrossstretchmarks]

*hello internet*

[@dharnie9861]

I was here 😁

[@KatherineSilzCarson]

Thank you for being here!

[@adamlv1]

The Prisoner’s Dilemma has a dominant strategy of always confessing. This was definitely the idea of someone who has never snitched on anyone or worked with police to get someone else more time. The results of confessing but then getting stabbed in prison would not be a dominant strategy in the end, but I understand game theory as it relates to dominant strategy and dominated strategy.

[@ulvumustafayev1823]

Hi.Thank you for this excellent presentation.I have watched all the videos and find them super useful.Unfortunately, there 2 exercises about game theory that I can't solve.Pleaseee could you help me? Pleasee

[@KatherineSilzCarson]

Ulvi - thank you for watching. I'm willing to try to help if you send me your question. Just reply to my comment and I'll get it.

[@ulvumustafayev1823]

@@KatherineSilzCarson A). Consider a duopoly with differentiated products. The two firms are denoted 1 and 2, and their products q1 and q2 with prices p1 and p2, respectively. There is a representative consumer whose net utility from purchasing the two goods is U=q1+q2-1/2(q1 ^2 + q2^2)-yq1q2-p1q1-p2q2 The parameter y, which ranges in between 0 and 1, is the degree of product differentiation (products are perfect substitutes when y= 1, while they are independent when y= 0. Firms have symmetric and constant marginal cost c 1. 1. Derive the direct and indirect demand functions (i.e., quantities as function of prices, and prices as function of quantities) 2. Consider then the case of quantity competition (Cournot). Derive the best response functions and the Nash equilibrium, as a function of the parameters y and c. 3. Do the same for the case of price competition (Bertrand). 4. Compare the two equilibria. Which one is preferred by the firms? And by the consumer? B. Consider a sealed‐bid first‐price auction. There are two bidders, 1 and 2, with valuations for the good of v1 and v2, respectively. Each buyer’s valuation is his/her own private information. v1 is uniformly distributed on include (0,1) and v2 is uniformly distributed on include (0,2). So player 1, for instance, knows v1 exactly but regarding v2 he/she knows only that it is uniformly distributed between 0 and 2. 1. Define the Bayesian equilibrium for this game. 2. Find a Bayesian equilibrium. 3. Discuss uniqueness.

[@ulvumustafayev1823]

@@KatherineSilzCarson If you don't understand my writing, I can send it to your mail if you give me. Thankss so much dear teacher.

[@KatherineSilzCarson]

@@ulvumustafayev1823 , for the first problem, first you need to set up and solve the consumer's utility maximization problem to get their demand and inverse demand functions. Then, you would use one of those functions to solve for the Cournot solution. Remember that in a Cournot duopoly, both firms simultaneously choose their quantities. You can see how to set up and solve a Cournot duopoly problem in my "Oligopoly" video. In Bertrand competition, both firms choose their price simultaneously. In a typical Bertrand equilibrium, firms end up choosing the same market price as would occur in perfect competition. However, in this example, your solution for the price should be a function of y, the degree of product differentiation. For the auction problem, I would encourage you to think about possible bid combinations for each player to find the equilibrium. Remember that in any equilibrium, neither player has an incentive to change their strategy. In a Bayesian equilibrium, a player's strategy consists of two parts: (1) their beliefs and (2) their plan of action. So, in equilibrium, neither player has an incentive to change either their beliefs or their action plan. Hope these hints help.

[@ulvumustafayev1823]

@@KatherineSilzCarson Thankssss so much for giving feedback👍👍

[@fatinnoor2422]

I like ur voice

[@KatherineSilzCarson]

Thank you! And thank you for watching!

[@channelhasbeenmoved639]

HELLO INTERNET WELCOME T-

[@legendaryzeko3601]

im dying bro😂😂😂😂

[@120201atta]

How do you know HOW each action earned them what point? Are u just making it up for example sake? WTF DOES those numbers MEAN?

[@KatherineSilzCarson]

Great question! Really, all the numbers tell you are how each player rank orders each of the possible outcomes in the game. Larger numbers mean better outcomes. Any set of numbers that generates the same rank ordering for a player are equally good. So in this sense, the numbers are definitely made up. However, for a real-world game, it is definitely possible to apply some reasoning to why a player might prefer Outcome A to Outcome B. The numbers that one uses for the payoffs from A and B are completely arbitrary, as long as the payoff from A is larger than the payoff from B. Hope this helps!

[@120201atta]

@@KatherineSilzCarson Katherine thanks for the prompt response to a question asked in frustration. So far I haven't managed to find a game theory/prisoner dilemma/nash equilibrium explanation for dummies that I can understand. All I'm looking for looks something like this... Hi today we're looking at game theory and how it's applied to businesses, who use it to justify financial strategies against competitors. It can be represented in a table. Here's a real life example with actual figures. This is what these two companies did and this is how you can do it too, check it out... Draw a table like so, add company A and Company B here and here like so. Add this number here if this and add that number there if that. *Then proceeds to fill in each cell with a number - not examples of numbers - while explaining exactly what they represent* That's all I'm looking for really.

[@120201atta]

​@@KatherineSilzCarson I struggle to follow your top/bottom left/right analogy, and how does top equal 1? What does top even mean in this scenario? Everyone in the comments gets it and seems to understand perfectly, what am I missing? Im not following the logic of anything yet everything sounds so well expained..

[@KatherineSilzCarson]

@@120201atta Try to think of it this way. Suppose that there are two business who are competing in the same market. They can either collude to keep prices high, or undercut each other. If one firm undercuts the other, then the firm that undercuts gets a larger market share, and thus, more profits. If they both undercut, their profits would be small. This is analogous to the prisoner's dilemma game. Make the row player firm 1, and the column player firm 2. Top, Left might be both colluding, with each player getting profits of 4. Top Right would be firm 1 colluding and firm 2 undercutting. In that case, firm 1's profits might be zero (because they lose their market share), and firm 2's might be 6 (because they get the whole market). Bottom Left would be the same, except the firms' actions and payoffs are reversed. Bottom right has both firms undercutting, in which perhaps they each make profits of 1. Each firm has a dominant strategy to undercut, even though if they could figure out a way to work together and collude, total profits to both firms would be higher. At that outcome, though, each still has an incentive to undercut their opponent and try to get 6. This happens in real life with OPEC sometimes (or with non-member countries reneging on their agreements with OPEC). Hope this helps put some more business context on the prisoner's dilemma for you. Thank you for your questions and for watching!