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The dollar auction is a non-zero sum sequential game designed by economist Martin Shubik to illustrate a paradox brought about by traditional rational choice theory in which players are compelled to make an ultimately irrational decision based completely on a sequence of apparently rational choices made throughout the game.
The setup involves an auctioneer who volunteers to auction off a dollar bill with the following rule: the bill goes to the winner; however, the second-highest bidder also loses the amount that they bid, making them the biggest loser in the auction. The winner can get a dollar for a mere 5 cents (the minimum bid), but only if no one else enters into the bidding war.
The game begins with one of the players bidding 5 cents, hoping to make a 95-cent profit. They can be outbid by another player bidding 10 cents, as a 90-cent profit is still desirable. Similarly, another bidder may bid 15 cents, making an 85-cent profit. Meanwhile, the second bidder may attempt to convert their loss of 10 cents into a gain of 80 cents by bidding 20 cents, and so on. Every player has a choice of either paying for nothing or bidding 5 cents more on the dollar. Any bid beyond the value of a dollar is a loss for all bidders alike.
A series of rational bids will reach and ultimately surpass one dollar as the bidders seek to minimize their losses. If the first bidder bids 95 cents, and the second bidder bids one dollar (for no net gain or loss), the first bidder stands to lose 95 cents unless they bid $1.05, in which case they rationally bid more than the value of the item for sale (the dollar) in order to reduce their losses to only 5 cents.
Bidding continues with the second highest bidder always losing more than the highest bidder and therefore always trying to become the high bidder. Only the auctioneer gets to profit in the end.
The game is analogous to the war of attrition and penny auction and has a symmetric mixed strategy equilibrium (there are also asymmetric pure equilibria). Suppose we start with two players; player 1 moves in odd periods, while player 2 moves in even periods. When a player is behind, they are indifferent between raising and dropping out. If the opponent drops out with probability , raising gives the player an expected payoff of On the other hand, dropping out yields zero, so indifference requires that the opponent drops out with 5% probability each period. Notice that this probability is independent of how high the bids currently are; this follows from the fact that past bids are a sunk cost. The maximum bid follows a binomial distribution with mean 0.5 (recall the bidders make zero profits each, on average).
In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally.
An auction is usually a process of buying and selling goods or services by offering them up for bid, taking bids, and then selling the item to the highest bidder or buying the item from the lowest bidder. Some exceptions to this definition exist and are described in the section about different types. The branch of economic theory dealing with auction types and participants' behavior in auctions is called auction theory.
The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the outcome is ideal for one player to yield, but the individuals try to avoid it out of pride for not wanting to look like a 'chicken'. So each player taunts the other to increase the risk of shame in yielding. However, when one player yields, the conflict is avoided, and the game is for the most part over.
In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (gambles)—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. The introduction of St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis. This hypothesis has proven useful to explain some popular choices that seem to contradict the expected value criterion, such as occur in the contexts of gambling and insurance.
A Vickrey auction 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.
An English auction is an open-outcry ascending dynamic auction. It proceeds as follows.
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by first considering the last time a decision might be made and choosing what to do in any situation at that time. Using this information, one can then determine what to do at the second-to-last time of decision. This process continues backwards until one has determined the best action for every possible situation at every point in time. It was first used by Zermelo in 1913, to prove that chess has pure optimal strategies.
In common valueauctions the value of the item for sale is identical amongst bidders, but bidders have different information about the item's value. This stands in contrast to a private value auction where each bidder's private valuation of the item is different and independent of peers' valuations.
In game theory, the war of attrition is a dynamic timing game in which players choose a time to stop, and fundamentally trade off the strategic gains from outlasting other players and the real costs expended with the passage of time. Its precise opposite is the pre-emption game, in which players elect a time to stop, and fundamentally trade off the strategic costs from outlasting other players and the real gains occasioned by the passage of time. The model was originally formulated by John Maynard Smith; a mixed evolutionarily stable strategy (ESS) was determined by Bishop & Cannings. An example is an all-pay auction, in which the prize goes to the player with the highest bid and each player pays the loser's low bid.
A unique bid auction is a type of strategy game related to traditional auctions where the winner is usually the individual with the lowest unique bid, although less commonly the auction rules may specify that the highest unique bid is the winner. Unique bid auctions are often used as a form of competition and strategy game where bidders pay a fee to make a bid, or may have to pay a subscription fee in order to be able to participate.
A bidding fee auction, also called a penny auction, is a type of all-pay auction in which all participants must pay a non-refundable fee to place each small incremental bid. The auction is extended each time a new bid is placed, typically by 10 to 20 seconds. Without new bids the last participant to have placed a bid wins the item and also pays the final bid price. The auctioneer makes money in two ways: the fees for each bid and the payment for the winning bid, totalling typically significantly more than the value of the item. Such auctions are typically held over the Internet, rather than in person.
A double auction is a process of buying and selling goods with multiple sellers and multiple buyers. Potential buyers submit their bids and potential sellers submit their ask prices to the market institution, and then the market institution chooses some price p that clears the market: all the sellers who asked less than p sell and all buyers who bid more than p buy at this price p. Buyers and sellers that bid or ask for exactly p are also included. A common example of a double auction is stock exchange.
Auction theory is an applied branch of economics which deals with how people act in auction markets and researches the properties of auction markets. There are many possible designs for an auction and typical issues studied by auction theorists include the efficiency of a given auction design, optimal and equilibrium bidding strategies, and revenue comparison. Auction theory is also used as a tool to inform the design of real-world auctions; most notably auctions for the privatization of public-sector companies or the sale of licenses for use of the electromagnetic spectrum.
A multiunit auction is an auction in which several homogeneous items are sold. The units can be sold each at the same price or at different prices.
A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids, so that no bidder knows the bid of any other participant. The highest bidder pays the price they submitted.
In economics and game theory, an all-pay auction is an auction in which every bidder must pay regardless of whether they win the prize, which is awarded to the highest bidder as in a conventional auction.
Revenue equivalence is a concept in auction theory that states that given certain conditions, any mechanism that results in the same outcomes also has the same expected revenue.
The generalized second-price auction (GSP) is a non-truthful auction mechanism for multiple items. Each bidder places a bid. The highest bidder gets the first slot, the second-highest, the second slot and so on, but the highest bidder pays the price bid by the second-highest bidder, the second-highest pays the price bid by the third-highest, and so on. First conceived as a natural extension of the Vickrey auction, it conserves some of the desirable properties of the Vickrey auction. It is used mainly in the context of keyword auctions, where sponsored search slots are sold on an auction basis. The first analyses of GSP are in the economics literature by Edelman, Ostrovsky, and Schwarz and by Varian. It is used by Google's AdWords technology, and it was employed by Facebook, which has now switched to Vickrey–Clarke–Groves auction
Solo 66 is a trick-taking, Ace-Ten, card game for five players in which a soloist always plays against the other four. It is based on the rules of Germany's national game, Skat, and is played with a French-suited Skat pack of 32 cards. Bidding is for the trump suit. Jacks are ranked within their respective suits and do not form additional trumps over and above the cards of the trump suit. Grupp describes it as "an entertaining game for a larger group."