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Topic: **Game theory**

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# ๐ Braessโs paradox

**Braess' paradox** is the observation that adding one or more roads to a road network can slow down overall traffic flow through it. The paradox was postulated in 1968 by German mathematician Dietrich Braess, who noticed that adding a road to a particular congested road traffic network would increase overall journey time.

The paradox may have analogies in electrical power grids and biological systems. It has been suggested that in theory, the improvement of a malfunctioning network could be accomplished by removing certain parts of it. The paradox has been used to explain instances of improved traffic flow when existing major roads are closed.

### Discussed on

- "Braess's Paradox" | 2021-05-16 | 64 Upvotes 30 Comments
- "Braessโs paradox" | 2018-09-22 | 134 Upvotes 37 Comments
- "Braessโ paradox" | 2017-01-08 | 136 Upvotes 91 Comments
- "Braess' paradox: adding a new road to a city can slow down traffic" | 2015-10-16 | 98 Upvotes 61 Comments

# ๐ Monty Hall Problem

The **Monty Hall problem** is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show *Let's Make a Deal* and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the *American Statistician* in 1975 (Selvin 1975a), (Selvin 1975b). It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in *Parade* magazine in 1990 (vos Savant 1990a):

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Vos Savant's response was that the contestant should switch to the other door (vos Savant 1990a). Under the standard assumptions, contestants who switch have a 2/3 chance of winning the car, while contestants who stick to their initial choice have only a 1/3 chance.

The given probabilities depend on specific assumptions about how the host and contestant choose their doors. A key insight is that, under these standard conditions, there is more information about doors 2 and 3 than was available at the beginning of the game when door 1 was chosen by the player: the host's deliberate action adds value to the door he did not choose to eliminate, but not to the one chosen by the contestant originally. Another insight is that switching doors is a different action than choosing between the two remaining doors at random, as the first action uses the previous information and the latter does not. Other possible behaviors than the one described can reveal different additional information, or none at all, and yield different probabilities. Yet another insight is that your chance of winning by switching doors is directly related to your chance of choosing the winning door in the first place: if you choose the correct door on your first try, then switching loses; if you choose a wrong door on your first try, then switching wins; your chance of choosing the correct door on your first try is 1/3, and the chance of choosing a wrong door is 2/3.

Many readers of vos Savant's column refused to believe switching is beneficial despite her explanation. After the problem appeared in *Parade*, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erdลs, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating vos Savantโs predicted result (Vazsonyi 1999).

The problem is a paradox of the *veridical* type, because the correct choice (that one should switch doors) is so counterintuitive it can seem absurd, but is nevertheless demonstrably true. The Monty Hall problem is mathematically closely related to the earlier Three Prisoners problem and to the much older Bertrand's box paradox.

### Discussed on

- "Monty Hall Problem" | 2022-06-09 | 24 Upvotes 116 Comments
- "Monty Hall Problem" | 2019-10-24 | 122 Upvotes 252 Comments
- "Monty Hall problem" | 2010-02-22 | 14 Upvotes 27 Comments

# ๐ Pirate Game

The **pirate game** is a simple mathematical game. It is a multi-player version of the ultimatum game.

### Discussed on

- "The pirate game" | 2016-01-12 | 298 Upvotes 135 Comments

# ๐ Sprouts is a pencil-and-paper game with interesting mathematical properties

**Sprouts** is a paper-and-pencil game that can be enjoyed simply by both adults and children. Yet it also can be analyzed for its significant mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in the early 1960s. Setup is even simpler than the popular Dots and Boxes game, but game-play develops much more artistically and organically.

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- "Sprouts is a pencil-and-paper game with interesting mathematical properties" | 2012-04-01 | 169 Upvotes 9 Comments
- "Sprouts game" | 2008-09-01 | 64 Upvotes 16 Comments

# ๐ The Price of Anarchy

The **Price of Anarchy** (**PoA**) is a concept in economics and game theory that measures how the efficiency of a system degrades due to selfish behavior of its agents. It is a general notion that can be extended to diverse systems and notions of efficiency. For example, consider the system of transportation of a city and many agents trying to go from some initial location to a destination. Let efficiency in this case mean the average time for an agent to reach the destination. In the 'centralized' solution, a central authority can tell each agent which path to take in order to minimize the average travel time. In the 'decentralized' version, each agent chooses its own path. The Price of Anarchy measures the ratio between average travel time in the two cases.

Usually the system is modeled as a game and the efficiency is some function of the outcomes (e.g. maximum delay in a network, congestion in a transportation system, social welfare in an auction, ...). Different concepts of equilibrium can be used to model the selfish behavior of the agents, among which the most common is the Nash equilibrium. Different flavors of Nash equilibrium lead to variations of the notion of Price of Anarchy as **Pure Price of Anarchy** (for deterministic equilibria), **Mixed Price of Anarchy** (for randomized equilibria), and **BayesโNash Price of Anarchy** (for games with incomplete information). Solution concepts other than Nash equilibrium lead to variations such as the **Price of Sinking**.

The term Price of Anarchy was first used by Elias Koutsoupias and Christos Papadimitriou, but the idea of measuring inefficiency of equilibrium is older. The concept in its current form was designed to be the analogue of the 'approximation ratio' in an approximation algorithm or the 'competitive ratio' in an online algorithm. This is in the context of the current trend of analyzing games using algorithmic lenses (algorithmic game theory).

### Discussed on

- "The Price of Anarchy" | 2017-01-09 | 120 Upvotes 135 Comments

# ๐ Parrondo's paradox

**Parrondo's paradox**, a paradox in game theory, has been described as: *A combination of losing strategies becomes a winning strategy*. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. A more explanatory description is:

*There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to construct a winning strategy by playing the games alternately.*

Parrondo devised the paradox in connection with his analysis of the Brownian ratchet, a thought experiment about a machine that can purportedly extract energy from random heat motions popularized by physicist Richard Feynman. However, the paradox disappears when rigorously analyzed. Winning strategies consisting of a combinations of losing strategies have been explored in biology before Parrondo's paradox was published. More recently, problems in evolutionary biology and ecology have been modeled and explained in terms of the paradox.

### Discussed on

- "Parrondo's Paradox" | 2023-04-28 | 69 Upvotes 48 Comments
- "Parrondo's paradox" | 2011-11-16 | 41 Upvotes 7 Comments
- "Parrondo's paradox: A losing strategy that wins" | 2010-06-01 | 43 Upvotes 12 Comments

# ๐ Vickrey Auction

A **Vickrey auction** or **sealed-bid second-price auction** (**SBSPA**) is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest bid. This type of auction is strategically similar to an English auction and gives bidders an incentive to bid their true value. The auction was first described academically by Columbia University professor William Vickrey in 1961 though it had been used by stamp collectors since 1893. In 1797 Johann Wolfgang von Goethe sold a manuscript using a sealed-bid, second-price auction.

Vickrey's original paper mainly considered auctions where only a single, indivisible good is being sold. The terms *Vickrey auction* and *second-price sealed-bid auction* are, in this case only, equivalent and used interchangeably. In the case of multiple identical goods, the bidders submit inverse demand curves and pay the opportunity cost.

Vickrey auctions are much studied in economic literature but uncommon in practice. Generalized variants of the Vickrey auction for multiunit auctions exist, such as the generalized second-price auction used in Google's and Yahoo!'s online advertisement programmes (not incentive compatible) and the VickreyโClarkeโGroves auction (incentive compatible).

### Discussed on

- "Vickrey Auction" | 2023-06-29 | 66 Upvotes 65 Comments

# ๐ Newcomb's paradox

In philosophy and mathematics, **Newcomb's paradox**, also referred to as **Newcomb's problem**, is a thought experiment involving a game between two players, one of whom is able to be able to predict the future.

Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore Laboratory. However, it was first analyzed in a philosophy paper by Robert Nozick in 1969, and appeared in the March 1973 issue of *Scientific American*, in Martin Gardner's "Mathematical Games." Today it is a much debated problem in the philosophical branch of decision theory.

### Discussed on

- "Newcomb's paradox" | 2015-04-03 | 54 Upvotes 67 Comments

# ๐ Herd Immunity

**Herd immunity** (also called **herd effect**, **community immunity**, **population immunity**, or **social immunity**) is a form of indirect protection from infectious disease that occurs when a large percentage of a population has become immune to an infection, whether through previous infections or vaccination, thereby providing a measure of protection for individuals who are not immune. In a population in which a large proportion of individuals possess immunity, such people being unlikely to contribute to disease transmission, chains of infection are more likely to be disrupted, which either stops or slows the spread of disease. The greater the proportion of immune individuals in a community, the smaller the probability that non-immune individuals will come into contact with an infectious individual, helping to shield non-immune individuals from infection.

Individuals can become immune by recovering from an earlier infection or through vaccination. Some individuals cannot become immune due to medical reasons, such as an immunodeficiency or immunosuppression, and in this group herd immunity is a crucial method of protection. Once a certain threshold has been reached, herd immunity gradually eliminates a disease from a population. This elimination, if achieved worldwide, may result in the permanent reduction in the number of infections to zero, called eradication. Herd immunity created via vaccination contributed to the eventual eradication of smallpox in 1977 and has contributed to the reduction of the frequencies of other diseases. Herd immunity does not apply to all diseases, just those that are contagious, meaning that they can be transmitted from one individual to another. Tetanus, for example, is infectious but not contagious, so herd immunity does not apply.

The term "herd immunity" was first used in 1923. It was recognized as a naturally occurring phenomenon in the 1930s when it was observed that after a significant number of children had become immune to measles, the number of new infections temporarily decreased, including among susceptible children. Mass vaccination to induce herd immunity has since become common and proved successful in preventing the spread of many infectious diseases. Opposition to vaccination has posed a challenge to herd immunity, allowing preventable diseases to persist in or return to communities that have inadequate vaccination rates.

### Discussed on

- "Herd Immunity" | 2020-03-14 | 46 Upvotes 72 Comments

# ๐ VickreyโClarkeโGroves Auction

A **VickreyโClarkeโGroves (VCG) auction** is a type of sealed-bid auction of multiple items. Bidders submit bids that report their valuations for the items, without knowing the bids of the other bidders. The auction system assigns the items in a socially optimal manner: it charges each individual the harm they cause to other bidders. It gives bidders an incentive to bid their true valuations, by ensuring that the optimal strategy for each bidder is to bid their true valuations of the items. It is a generalization of a Vickrey auction for multiple items.

The auction is named after William Vickrey, Edward H. Clarke, and Theodore Groves for their papers that successively generalized the idea.

The VCG auction is a specific use of the more general VCG Mechanism. While the VCG auction tries to make a socially optimal allocation of items, VCG mechanisms allow for the selection of a socially optimal outcome out of a set of possible outcomes. If collusion is likely to occur among bidders, the VCG outperforms the generalized second-price auction for both revenues produced for the seller and allocative efficiency.

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- "VickreyโClarkeโGroves Auction" | 2019-07-31 | 82 Upvotes 34 Comments