Best whatever happens, and still worse?

Two coffee shops on one street

Every Monday each coffee shop decides: keep its price or offer a discount. The owners can't make a deal. Weekly profit, in thousands:

Find the dominant strategy

The test is simple: forget your favourite option and compare column by column. Take the first column (the rival picked its first option) and see which of your rows gives you more. Then the second column. If the same row wins in both, it is dominant. For the rival it's the same, except you compare its numbers row by row. Try it in three games.

A dominant strategy is a luxury that turns up rarely. In "where to meet" the best choice depends entirely on the other person, and then agreements and habits matter. In the newcomer game something else is interesting: only the leader has a dominant strategy, but it's enough for the newcomer to know about it. If advertising always pays for the leader, the leader will advertise, and the newcomer responds to the advertising rather than to guesses. Someone else's dominant strategy is a clue too.

What a dominant strategy is

A dominant strategy is an option that gives a player more than any other option of theirs, whatever the other players choose. The opposite is a dominated strategy: there is another option that beats it whatever the others choose. A rational player never picks a dominated strategy, and always picks the dominant one if there is one.

• Columns, not the biggest numberA common mistake is to pick the row with the biggest number in the table. Dominance is checked by comparing within each column separately: your option has to win in all of them.
• Strict and weakA strictly dominant strategy is better everywhere. A weakly dominant one is never worse and better somewhere. An honest bid in a second-price auction is weakly dominant: it often gives the same as other bids, but never less.
• Best for you ≠ best togetherDominance is about one player with the others held fixed. It promises nothing good about what happens when everyone plays their dominant strategy, as the price war shows.
• Knowing it is enoughTo play a dominant strategy you need no assumptions about the rival and no faith in their rationality. That is why "the player will play the dominant strategy" is the most reliable prediction in game theory.

Cross out what can't win

Here each player has three options, and neither has a dominant strategy. But some options are dominated: another option beats them whatever the rival picks. A rational player won't pick them, and the rival, knowing that, can cross them out and look at a smaller table. In the smaller table new dominated options appear. Each cell shows your payoff first (rows), then the rival's (columns). The "5 · 5" cell is the best for both. Will rational players get there? Tap a row or a column you think can be crossed out.

Each step here rests on the one before: the rival crosses out a column only if it believes you won't pick a row that is already crossed out, and you only if you believe the rival reasons the same way. The longer the chain, the more faith in the other's rationality it needs, and the less reliable the prediction when real people play.

Where dominant strategies live

• Price warsA discount that steals the rival's customers often pays whatever the rival does. That's why markets with a few similar players slide so easily into a price war in which all the sellers lose and the buyers win.
• A lowest-price guarantee"Find it cheaper and we'll refund the difference" sounds like a gift to buyers, but it actually changes the table. If both coffee shops match each other's price, a cut no longer steals customers: it gives 1 instead of 3, and against the neighbour's cut the same 1. Keeping the price becomes a weakly dominant strategy.
• Advertising racesIf advertising lures customers from rivals, advertising pays for each one alone, but together everyone pays and nobody lures anyone. Avinash Dixit and Barry Nalebuff, in "Thinking Strategically" (1991), describe the 1971 ban on cigarette ads on US television as a favour to the tobacco companies themselves: it stopped a race none of them could leave alone.
• The second-price auctionThe highest bid wins, and the winner pays the second-highest. In 1961 William Vickrey showed that bidding exactly what the item is worth to you is weakly dominant here. Bidding higher sometimes makes you overpay; bidding lower sometimes loses you a profitable win. eBay's automatic bidding works almost the same way.

The formula and the link to equilibrium

Not beaten by any option, yet beaten by a mix

Weak dominance: the order matters

Where the idea came from

• Honesty as a strategyIn 1961, in a paper on auctions, William Vickrey first showed rules under which telling the truth is weakly dominant for every participant. In 1996 he received the Nobel memorial prize in economics for this and other work.
• Voting without tricks is impossibleAllan Gibbard (1973) and Mark Satterthwaite (1975) proved that if there are at least three candidates and each can win, then any voting rule under which an honest vote is weakly dominant for every voter is a dictatorship: one voter decides the result. Under every other rule, someone sometimes gains by not voting for their favourite.
• Dominance against the averageIn 1969 the philosopher Robert Nozick, in "Newcomb's Problem and Two Principles of Choice", described a problem where the rule "pick what dominates" and the calculation of the average payoff give different advice, and there is still no agreement on which is right.
• From crossing out to rationalizabilityIn 1984 Douglas Bernheim and David Pearce, in the same issue of Econometrica, independently described which strategies a rational player who knows the others are rational too may choose. For two players these are exactly the ones left after crossing out all strictly dominated strategies, including those dominated by a mix.