Game Theory for beginners

Understanding types of games

Like most millennials, I can’t stop playing games on my computer or phone. As a result, I have never been more eager to write about Game Theory (this is my 83rd post by the way).

But wait, what is Game Theory in the first place?

Before we understand Game Theory, we must understand what is a Game in terms of mathematics.

A Game is an ecosystem/situation/environment with rules & regulations, involving multiple stakeholders (players) where each player takes a certain decision/follow some strategy to reach an ultimate goal.

Sounds very mechanical. Right?

Consider your favorite game, say Ludo. Why do we call Ludo a game? because

• It has a certain set of rules & regulations to be followed like you can’t move diagonal, every player gets a chance to throw the dice in a round-robin fashion & not in an arbitrary manner, etc.
• It involves 2–4 players
• Each player has an ultimate aim, to cross all his/her 4 tokens across the board
• Each player takes a decision to move a particular token (out of the 4 tokens) once the dice are thrown.

Game theory is the mathematical way of choosing the best strategy/decision (for a particular player) of all possible options to reach to the ultimate goal given a particular situation in the game. You might be required to change your strategy midway in a game if the situation changes drastically.

For example: In T20 Cricket, if you don’t lose wickets early, the team may strategize to play aggressively while if the team loses 3 wickets in the 1st over, they may play defensively. Hence, changes in strategy depend on the situation.

As we are now clear with Game & Game Theory, let’s understand a few important concepts

Zero Sum games: Games where one player’s gain is the other player’s loss with the magnitude being the same for both cases. Hence Gains + Losses = 0.

For example: In the game of Gambling between 2 players, if player A wins 100 bucks, player B loses 100 bucks. Hence the magnitude for gain & loss (money in this example) remains the same in any condition. In some other scenario, If play B wins 200 bucks, then player A loses 200 bucks hence gains=loss in any condition.

Any reference to Gains + Loss in the rest of the blog means summation of gains made by each player + summation of loss made by each player

Constant Sum games: Similar to Zero-sum games, Constant sum games are games where Gains + Loss !=0 but is equal to some constant C for any scenario.

In the case of a game of lottery, if you win, you make 10000 bucks but no loss in case you didn’t win. So if 50 folks bought lottery tickets & only one of them won, then Gains + Loss !=0 i.e. 1x10000 + 49x0 = 10000!=0 but whosoever wins, the Gains + Loss = 10000 hence a constant sum irrespective of the outcome (even if different players win, Gain+loss remains constant)

Non-Zero-sum games: As the name suggests, games in which Gains + Loss are different for different scenarios/strategies are called Non-Zero sum games.

For example: In stone-paper-scissors,

• if player A/player B wins, he gets 100 bucks & other gets 0
• But if both players show up the same sign, both get 10 bucks

So, in this game, the gain + loss isn’t the same for all possible strategies.

• If players show up different sign, gain+loss = 100+0=100
• If players show up same size, gain + loss = 20+0 =20

Extensive form games

Games that require the players to take multiple decisions in a sequence to reach the ultimate goal are called extensive form games. Such games can be represented by a tree diagram as below:

In the above diagram, The player starts from state 1 and can take 3 decisions (Left, Right, or Jump), on reaching state 2, the player can take 2 decisions (go, stop) leading to the final position.

Perfect information games

Games in which

• The current configuration of the game is known to each player before taking any decision
• If a player takes a decision, the resultant configuration is also known

For example: In any board game (assume Snakes & Ladders), each player knows the game’s config beforehand (i.e. where are tokens of other players on the board, where is their token) before taking the decision & once he takes a decision (rolls the dice), the resultant config is also known to him.

So if a player is at 75 in snakes & ladders, & dice show up at 5, he moves to 80 (this is guaranteed, no uncertainty in the final config of the game)

Incomplete information games

Opposite to Perfect Information games, in such games, the entire game config is not known beforehand & the player might need to take a decision without knowing the position of other players in the game. Also, he isn’t aware of what will be the outcome of his decision in the game.

For example: During a penalty shootout, the goalie takes a decision (dives to the right or left) based on intuition without knowing in which direction the striker will kick the ball. Here, the goalie doesn’t know where the Striker will kick the ball (unknown of other player’s strategy and position) & the result is also unknown to him when he takes the action.

Cooperative games:

Multiplayer games (more than 2) where individuals are benefitted when teaming up with other players compared to playing alone are called Cooperative games. Elections can be a fitting example of where parties move into coalitions to form governments.

This should be enough for the day, I will continue next week by taking a deeper dive into game theory & related concepts. Till then