[Incoherent Rambling] Optimal strategy in game theory - Codeforces (2024)

To whom it may concern,

Today I was reading some recent problems when I stumbled upon this div2B. 1956B - Nene and the Card Game. At first, it seems like an innocent game theory problem, until I read the following sentence.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

This struck me as a bit odd, because in most game theory problems on Codeforces, it's a win/lose/draw game, or the player wants to maximize the difference between their score and their opponent's score, or they want to maximize their own score and all outcomes have the same sum of scores for the two opponents (which is equivalent). But Nene's objective is most peculiar, where she doesn't care about how many points her opponent has at all, until she cannot get any more for herself and then it is the only thing she cares about.

Now, what does the problem say is the strategy of Nene's opponent (you)?

...what is the maximum number of points you can get by taking your turns optimally?

Hmm, it seems that you do not care about Nene's score, although she cares about your score. Oh well.

At this moment I thought, "is this problem even well-defined?" What does an optimal strategy mean in this game? We seem to take it for granted that it even makes sense to talk about an "optimal strategy" at all. There are a few possible concepts of what an optimal strategy could mean. Maybe it means playing the move that gives you the best guaranteed score, assuming the worst case for your opponent's decisions. Or maybe it means playing the move that gives you the best outcome assuming that your opponent is rational and plays optimally. Yet another interpretation could involve Alice and Bob agreeing to collaborate, if they see a path that makes them both better-off than if they competed.

In most game theory problems on Codeforces, we are dealing with a two-player, zero-sum (or constant-sum), turn-based, finite, perfect-information game. In these cases, multiple interpretations of optimal play turn out to be equivalent. The one where you try to optimize your guaranteed score is called Maximin and the one where you try to optimize your score assuming a rational opponent is called Minimax. Both of these solution concepts are equivalent for this class of games.

However the idea of "maximize your score first, then minimize your opponent's score second" seems to imply that we are outside of zero-sum territory (otherwise maximizing your own score would be enough). We cannot guarantee that Minimax = Maximin in non-zero-sum games. For example, consider this simple example game.

Alice goes first.Choice 1: Alice and Bob both get 1 point, game ends.Choice 2: Go to BobIf Bob has a turn:Choice 1: Alice gets 0 points, Bob gets 1 point, game ends.Choice 2: Alice gets 2 points, Bob gets 2 points, game ends.
  • Maximin: Suppose Alice wants to maximize her guaranteed score, assuming the worst case. Then she should make choice 1, because she can guarantee 1 point, while the worst case in choice 2 is 0 points.
  • Minimax: Suppose Alice wants to maximize her score assuming that Bob is rational and will try to maximize his own score. Then she should go with choice 2, since Bob will rationally get 2 points for each of them, which is better than 1 point.

In this example, we see how non-zero-sum games can result in different optimal strategies depending on which convention we choose. In general, the problem statement would need to be clear about which convention we are using.

So, what about this div2B problem? Is it well-defined? Technically, yes it is. After analyzing the mechanics of the game, you will notice that the sum of scores of the two players is the same in all outcomes. In fact, for each type of card, one of them will be played first and the other will be played second, resulting in $$$1$$$ point for one of the players. So the sum of scores will always be $$$n$$$ in every outcome, meaning it's a constant-sum game in disguise.

So, what lesson can we take from my incoherent rambling? Mainly, optimal play in game theory is surprisingly nuanced. In my opinion, problems on Codeforces should be immediately clear that they are well-defined, and this div2B problem requires a key observation in order to realize that it is well-defined. Maybe when we have zero-sum games in the future, we should make it clear that it is zero-sum, even if it's just a statement like "We can prove that the sum of the players' scores is the same in all outcomes", and maybe linking to a resource about the definition of optimal play in zero-sum games.

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[Incoherent Rambling] Optimal strategy in game theory - Codeforces (2024)


What is the optimal strategy for a game theory? ›

An optimal strategy is one that provides the best payoff for a player in a game. Optimal Strategy = A strategy that maximizes a player's expected payoff. Games are of two types: cooperative and noncooperative games.

How to approach game theory problems? ›

Methods of solving such a problem:
  1. Odds method ( 2 × 2 game without saddle point)
  2. Dominance method.
  3. Sub-games method – (for m × 2 or 2 × n matrices)
  4. Graphical method.

What is game theory in CP? ›

Game Theory is a topic in competitive programming that involves a certain type of problem, where there are some players who play a game based on given rules and the task is often to find the winner or the winning moves.

What are pure and mixed strategies in game theory? ›

Pure Strategy: Players choose a strategy to select a single action and play it - so far we have considered this scenario. Mixed Strategy: Players randomize over the set of available actions according to some probability distribution - a player randomizes and mixes between different actions.

What is a strongly dominant strategy in game theory? ›

Summary. The dominant strategy in game theory refers to a situation where one player has a superior tactic regardless of how the other players act. The Nash Equilibrium is an optimal state of the game, where each opponent makes optimal moves while considering the other player's optimal strategies.

How to calculate the optimal strategy? ›

The optimal strategy for the column player is to set the probability of playing Column 1 equal to q = d − b a − b − c + d The column player's probability of playing Column 2 is then determined as 1 − q. ν = ad − bc a − b − c + d .

What is the most famous example of game theory? ›

The most famous game is the prisoneris dilemma. A dominant strategy is a strategy that produces a higher payoff than any other possible strategy. No matter what your opponent might do, you play the dominant strategy. 11 / 64 Page 12 Static Games In the prisoneris dilemma, betray is a dominant strategy for both players.

What are the two strategies of game theory? ›

Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy.

What is the rule of dominance in game theory? ›

The principle of dominance states that if one strategy of a player dominates over the other strategy in all conditions then the later strategy can be ignored. A strategy dominates over the other only if it is preferable over other in all conditions.

What are the three basics of game theory? ›

The three basic elements of any game are: A set of participants, or "players." The moves, or "actions," that each player may make. The scores, or "payoffs," that each player earns at the end of the game.

What are the four types of games in game theory? ›

Cooperative and non-cooperative, symmetric and asymmetric, simultaneous and sequential are some of the different types of game theory.

What is game theory for beginners? ›

A key aspect of game theory is that the payoffs to one player depend on the decision that they make and on the decision that other players make. This means that in order for individuals to maximize their own payoffs, they must consider and predict what the other player(s) in the game will do as well.

What are the 4 rules of game theory? ›

There are 4 main elements in game theory:
  • Player: The strategic decision-maker.
  • Strategy: The rules that apply in the specific game.
  • Outcome: The result after a decision has been made.
  • Equilibrium: The point in a game where both players have made their decisions and can't make any other move.
Jun 10, 2021

What is optimal strategy in game theory? ›

game theory

…can deduce strategies that are optimal, which makes the outcome preordained (strictly determined). In chess, for example, exactly one of three outcomes must occur if the players make optimal choices: (1) White wins (has a strategy that wins against any strategy of Black); (2) Black wins; or (3) White and…

What is the Maximin strategy in game theory? ›

The Maximin Strategy is a conservative decision-making technique used in game theory, statistics, and philosophy. It aims to achieve the 'best of the worst' possible outcome in a decision scenario by maximizing the minimum gain.

What is an optimal decision in game theory? ›

The intention of game theory is to produce optimal decision-making of independent and competing actors in a strategic setting. Using game theory, real-world scenarios for such situations as pricing competition and product releases (and many more) can be laid out and their outcomes predicted.

What is optimal game theory? ›

Game theory optimal, or GTO poker strategy, is a strategy that seeks complete balance in the game, making your plays 100% unexploitable by your opponents. This style of poker is the exact opposite of the exploitative poker strategy, which most players from the older generations employ.

What is an efficient strategy in game theory? ›

An outcome is Pareto efficient if there is no other outcome that increases at least one player's payoff without decreasing anyone else's. Likewise, an outcome is Pareto inefficient if another outcome increases at least one player's payoff without decreasing anyone else's.

What is the optimal outcome in game theory? ›

An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. That is, a Pareto Optimal outcome cannot be improved upon without hurting at least one player.

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