We have 3 points and a game: we could bet n (n>0, n is real number) from our points. With probability of 0.4 we get 2n, otherwise lose it. The goal is to get 8 points. What's the probability and the optimal strategy for winning?

The expectation is negative, so, we'll assume that the chances are higher with less games. Firstly, let's put 3, lose or get 6. Then put 2, win or have 4. Put 4, lose or win. So, the probability of winning is 0.256. Is this strategy optimal and how could I prove it or get a better one?

As mentioned in the title. The proof (open source. found from MIT) goes as follow:

https://preview.redd.it/pkzejhnjg0yd1.png?width=2038&format=png&auto=webp&s=356582b049b5c82c57d9e786aea7ba1441086c08

However, I don't really understand case 2 and 3.

For case 2, All trees are -+ which means a win for Richard, then why Louise(+-) has a winning strategy. Is it a typo or my understanding issue.

For case 3, Richard's best strategy is a drawing strategy, and thus it guarantees he will end in a draw, so L must be a drawing strategy for Louise. Is my understanding correct for this case?

Thanks!

https://preview.redd.it/ausp533i9yxd1.png?width=488&format=png&auto=webp&s=c8a61032bd9a20df79aa148391a48d4cdf2e0a20

https://preview.redd.it/be2fg1oj9yxd1.png?width=472&format=png&auto=webp&s=19ae5bb90c442326b78ad8a6cec6143db7e92914

I'm creating a general game theory page on nonzerosum.games and wondered if you thought there were other important aspects of game theory that should be included on this page.

My supervisor referred to a book on Game Theory but forgot the name. It was supposedly written by a lifelong practitioner who applied Game Theory to his own life (may be 50 years back?) and all his life situations (including with his kids) and the lessons are applicable in business and competition.
Sounded interesting. He shared with me knowing that I like reading books.

I was hoping the experts here can help me find the book.

Imagine a basic scenario:

The Government needs a new power plant built. Their goal is to get the power plant built to specification in time for the lowest price, thus saving taxpayer money.

They open the floor for Contractors to bid, following whatever method the Government prefers (open bidding, blind bidding, etc). Their goal is to be awarded the contract and maximize their profit.

At a glance these two goals work in opposition--the Government wants to drive bid price down, and the Contractors want to keep bid price high. In theory, the Government induces competition among Contractors, keeping costs low and incentivizing winning Contractors to be efficient with their time, budget, and materials. However, as observed in reality, Contractors have indirect methods of achieving their win condition.

A basic example of this is to bid as low as reasonably possible to secure the contract in order to lock competitors out, then to operate with little concern for efficiency. Since the power plant is critical infrastructure that the Government needs built, the Contractor has leverage over the Government. They can, in effect, hold the project hostage by saying that if additional time and/or budget are not allocated, the project will not be finished. They are instead incentivized to operate slowly and inefficiently with materials and budget so that they are able to induce a budget expansion.

The Government does have the option to, instead of approving a budget expansion, to terminate the existing Contractor and hire the second best Contractor to finish the job. However, this still means that the full budget for this project was already allocated and the power plant is still not built to specification. It also means that Contractors are able to get paid in full despite not performing their full duties, and simply puts another Contractor in the same position of power as the first, with all the same incentives in place.

In reality, many variables--capital, material, labor, political, emotional, environmental, and more--go into determining the precise outcome of any given project. However, I am wondering what Game Theory has to say about this very basic example.

Is bidding in this scenario a fundamentally flawed approximation of the real cost of the power plant? Is the push and pull between Government and Contractor a better system for approximating the real cost?

It seems in this scenario that the Government ultimately has less leverage than the Contractor. Short of violence (physical or political), what options does the Government realistically have (if any) to incentivize the Contractor to align with the Government's time and budget goals for the power plant while remaining within specification? If they strong-arm the Contractors too much, Contractors would be operating at a loss and could not complete the project, even in good faith.

i’ve been taught the definition of Weakly dominance as: strategy s' dominates s" if payoff is at least greater or equal or in some cases equal 2nd POINT: at least 1 strat for which you have something strictly greater

so technically a strict domination would also fit into this definition right? so all strict dominations are also weak dominations but not the other way around?

i found it to be w, y. z for player 2 and a, c, d for player 1. is this right? i’m still relatively new to this

“A strategy that, given the strategies played by all the other players, yields the highest payoff compared to all other strategies of the player”

i thought this definition was dominant strategy but now i’m thinking its best reply since it mentions “given the strat played by all other players.” thoughts?

How can I improve my life with this.

I tried out the newly popular game under the mode Liar's Deck - Basic, and I feel like there's lots of stuff to analyze using game theory and probability. At first there aren't a lot of clues as to if a player is lying or not, so people usually play it safe by passing. Towards the later stage of one round, the probability of playing a fake card increases because so many cards have already been played and it's always possible that among those contained some real ones. Especially if someone plays like 2 or 3 cards at a later stage, that's very suspicious.

But because of that thinking, players usually are greedy in the early stages hoping to not get noticed with playing fake cards and then leave the real cards towards the end to bait you to call them a liar.

Feels like there's lots of theory and probability hidden, and it'd be cool to see if there's a Nash in here. Psychological factors also exist such as the delay time in playing cards. If someone took a lot of time to think about what to play, then maybe that person is lying.

if i’m understanding correctly, no one has a dominant strategy here. then, is it that for strictly dominated strategies, it’s:

• bell strictly dominated (SR) by lapses
• bell SR by echoes
• moon SR by wall

is this correct? i’m not sure if i’m figuring it out correctly. it’s just whichever one row/column of the same color is less than the other right? any tips for understanding would be helpful!

A historical and philosophical lens of game theory has led me to formulate a rather pessimistic outlook: From very logical assumptions on rational decision-making, models consistently find that innefficiences in systems are inevitable. Flaws are inherent in theoretical models, despite refinements. The interaction between subjective and objective aspects can lead to dubious conclusions from reasonable assumptions and sound logic.

Game theory is our attempt at rationalizing nature, the very essence of science. It is worrying that the field appears to be fundamentally broken. I have been self-learning game theory for about a year. I know I am wrong, that the field is not broken, why?

I'm interested in learning about game theory and its relationship in the real world, so far the only "reputable" books from recommendations are textbooks. I don't have time to read 500+ pages.

The typical prisoner's dilemma makes it so if the other person cooperates, you're better off defecting because you go from (e.g.) 3 years in prison to 2. But what if you were better off cooperating if the other party defects, but better off defecting if your partner cooperates?

If I notate the typical problem as:

(1,1) (3,0)
(0,3) (2,2)

And the case I'm describing is

(1,1) (2,0)
(0,2) (3,3)

Locking the Y axis to the top row, the X axis is best choosing the right. But if I lock the Y axis to the bottom row, X axis is best choosing left.

I thought at first that the answer was simply "there is no Nash Equilibrium", but Wikipedia states "Nash showed that there is a Nash equilibrium, possibly in mixed strategies, for every finite game." How does one go about working out what the Nash equilibrium is in a case like this?

Does anyone know any examples of a Prisoner's Dilemma in a Tragedy of the Commons situation? Or any interesting articles related to that?

I've been reading up on papers on Search in imperfect information games.

It seems the main method is Subgame Resolving, where the game is modifed with an action for the opponent to opt out (I believe at the move right before the state we are currently in) at the value of the "blue print" strategy computed before the game started. Subgame Resolving is used in DeepStack and Student of Games.

Some other methods are Maxmargin Search and Reach Search, but they don't seem to be used in a lot of new papers / software.

ReBeL is the weird one. It seems to rely on "picking a random iteration and assume all players’ policies match the policies on that iteration." I can see how this should in expectation be equivalent to picking a random action from the average of all policies (though the authors seem nervous about it, saying "Since a random iteration is selected, we may select an early iteration in which the policy is poor.") However I don't understand how this solves the unsafe search problem.

The classical issue with assuming you know the range/distribution over the opponent cards when doing subgame CFR is that you might as well just converge to a one-hot strategy. Subgame Resolving "solves" this by setting a limit to how much you are allowed to exploit your opponent, but it's a bit of a hack.

I can see that in Rock Paper Scissors, say, if the subgame solve alternates between one-hot policies like "100% rock", "100% paper" and "100% scissors", stopping at a random iteration would be sufficient to be unexploitable. But how do we know that the subgame solve won't just converge to "100% rock"? This would still be an optimal play given the assumed knowledge of the opponent ranges.

All this makes me think that maybe ReBeL does use Subgame Resolving (with a modified gadget game to allow the opponent an opt out) after all? Or some other trick that I missed?

The ReBeL paper does state that "All past safe search approaches introduce constraints to the search algorithm. Those constraints hurt performance in practice compared to unsafe search and greatly complicate search, so they were never fully used in any competitive agent." which makes me think they aren't using any of those methods.

TLDR: Is ReBeL's subgame search really safe? And if so, is it just because of "random iteration selection" or are there more components to it?

Hello I am not a game theorist and don't have any knowledge related to the subject. But I was recently was doing some writing on civility politics and civic discourse my main conclusion is that the biggest issue with civic discourse is not a lack of civility but a lack of ideological consistency. To speak about this I came up with an analogy( I am 100% sure something like this would already exist within the field I don't think what I am about to say is novel ). Imagine that you were playing in a soccer game and the referee decided that each team would self regulate. In this situation most people would agree that the soccer game would be considerably worse. Players would not only be positively enforced to always make biased calls but they would be negatively enforced to make good calls. I am sure this is like some game theory 101 stuff but what concept in game theory am I hitting on so I could read more about this. I think that self regulating speech is far better option than governmental control but I think if we are to apply game theory to the real world ( as I know we should not) It seems hard to escape this loop with our own actions.

The title :)

Someone recently described game theory to me as "everything can be solved mathematically".

I nodded and said "I'm sure most things can be". They became terse.
"No, not most things. EVERYTHING".

Naturally, I was skeptical, but intrigued.

So, yes... what is game theory useful for? Where do I start?

This is a variant of the Rock-Paper-Scissors game between two players – players 1 and 2 – in which player 2 can only play R or P. The game is described as follows:

1 \ 2 R P
R 0, 0 -1, 1
P 1, -1 0, 0
S -1, 1 1, -1

When I worked out the Mixed Strategy Nash Equilibrium, I noted that they have equal probabilities of winning, which seems wrong intuitively.

https://preview.redd.it/1vcetpvj1wvd1.png?width=1205&format=png&auto=webp&s=d608961c04ec75ab94e77cb2108e4b1eb4704866

Analyse and find all the Nash Equilibria (including pure and mixed strategy NE) for the following game table. Explain why if there is none. (Note: You need to present in a clear and easy-to-understand manner.)

I understand that with best response analysis, you get 3 Nash Equilibrium (B,A), (B,B) and (C,C).

However, I also understand that the game is dominance solvable through the iterated elimination of strictly dominated strategies. Resulting pure Nash Equilibrium is (C,C)

Hence, I conclude that there is no mixed strategy nash equilibrium because a pure nash equilibrium exists.

But how should I prove this? How do I explain where there is no mixed strategy nash equilibrium?

How would you do this?

Pretend you are playing a game with 49 other players, 50 players total including you. You must all choose a number between 0-101 (1-100). The number you choose represents the number of points you get. However, if any of the players choose the same number they both receive 0 points. The player with the highest amount of points wins. What number would you choose and why?