Positional game

Key takeaways

Positional game belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Positional game to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Positional game from memory before moving on to harder problems.

Reference excerpt

A positional game in game theory is a kind of a combinatorial game for two players. It is described by:

X {\displaystyle X} – a finite set of elements. Often X {\displaystyle X} is called the board and its elements are called positions.

F {\displaystyle {\mathcal {F}}} – a family of subsets of X {\displaystyle X} . These subsets are usually called the winning sets. A criterion for winning the game. During the game, players alternately claim previously-unclaimed positions, until one of the players wins. If all positions in X {\displaystyle X} are taken while no player wins, the game is considered a draw. The classic example of a positional game is tic-tac-toe. In it, X {\displaystyle X} contains the 9 squares of the game-board, F {\displaystyle {\mathcal {F}}} contains the 8 lines that determine a victory (3 horizontal, 3 vertical and 2 diagonal), and the winning criterion is: the first player who holds an entire winning-set wins. Other examples of positional games are Hex and the Shannon switching game. For every positional game there are exactly three options: either the first player has a winning strategy, or the second player has a winning strategy, or both players have strategies to enforce a draw. The main question of interest in the study of these games is which of these three options holds in any particular game. A positional game is finite, deterministic and has perfect information; therefore, in theory it is possible to create the full game tree and determine which of these three options holds. In practice, however, the game-tree might be enormous. Therefore, positional games are usually analyzed via more sophisticated combinatorial techniques.

Alternative terminology Often, the input to a positional game is considered a hypergraph. In this case:

The elements of X {\displaystyle X} are called vertices (or points), and denoted by V; The elements of F {\displaystyle {\mathcal {F}}} are called edges (or hyperedges), and denoted by E or H.

Variants There are many variants of positional games, differing in their rules and their winning criteria.

Different winning criteria Strong positional game (also called Maker-Maker game) The first player to claim all of the elements of a winning set wins. If the game ends with all elements of the board claimed, but no player has claimed all elements of a winning set, it is a draw. An example is classic tic-tac-toe. Maker-Breaker game The two players are called Maker and Breaker. Maker wins by claiming all elements of a winning set. If the game ends with all elements of the board claimed, and Maker has not yet won, then Breaker wins. Draws are not possible. An example is the Shannon switching game. Avoider-Enforcer game The players are called Avoider and Enforcer. Enforcer wins if Avoider ever claims all of the elements of a winning set. If the game ends with all elements of the board claimed, and Avoider has not claimed a winning set, then Avoider wins. As in maker-breaker games, a draw is not possible. An example is Sim. Discrepancy game The players are called Balancer and Unbalancer. Balancer wins if he ensures that in all winning sets, each player has roughly half of the vertices. Otherwise Unbalancer wins. Scoring game Comparing at the end the winning sets obtained by the players, whoever has the winning set with the highest score wins, where the score of each winning set is given in the instance. An example is the Largest Connected Subgraph Game, where the positions are the vertices of a graph, the winning sets are connected subgraphs and the winner is the one who obtains the largest connected subgraph.

Different game rules Waiter-Client game (also called Picker-Chooser game) The players are called Waiter and Client. In each turn, Waiter picks two positions and shows them to Client, who can choose one of them. Biased positional game Each positional game has a biased variant, in which the first player can take p elements at a time and the second player can take q elements at a time (in the unbiased variant, p=q=1).

Specific games The following table lists some specific positional games that were widely studied in the literature.

See also Topological game, a generalization of a positional game to infinite sets Banach–Mazur game, a game played on a topological space by choosing among certain subsets, with winning conditions resembling those of a maker-breaker game

References

Worked examples

Example 1 — a first encounter with Positional game

Start with the simplest possible case. Write down what Positional game claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Positional game before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Positional game ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Positional game

In research: Positional game appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Positional game in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Positional game is common in secondary-school and first-year university syllabi. It links to neighbouring topics Positional games, so understanding it makes those chapters shorter.
In everyday life: Look for Positional game outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Positional game in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Positional game means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Positional game out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Positional game in simple terms?

A positional game in game theory is a kind of a combinatorial game for two players. It is described by: X {\displaystyle X} – a finite set of elements.

Why does Positional game matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Positional game?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Positional game.