Introduction to Cognition and Gaming 9/25/02: Von Neumann’s Game Theory, Game Balance

John von Neumann Taught at Princeton University during the 1950’s Colleague of Kurt Gödel Colleague of Albert Einstein Students called von Neumann “The Genius”

The Theory It is always possible to find an equilibrium from which neither player should deviate unilaterally in any game that satisfies the following criteria:  The game is finite – both in number of options at each move, and in total number of moves to the end of the game  The game is zero-sum – one player’s gain is exactly the other’s loss  The game is one of complete information – each player knows all options available to her and to her opponent, as well as outcome values and scale of values

Cutting the Cake Choose bigger piece Choose smaller piece Cut cake as evenly as possible Half the cake minus a crumb Half the cake plus a crumb Cut one piece bigger than the other Small pieceBig piece Cutter Strategy Chooser Strategy

Finding the Saddle Point

Rock-Paper-Scissors Simple, symmetric two-player game Rock defeats scissors Scissors defeat paper Paper defeats rock Same item results in a tie Von Neumann equilibrium – play at random with probability 1/3 RPS R01 P10 S 10

John Nash Generalized coalition-free games for several players Nash equilibrium is the strategy that results in all parties being satisfied by playing the strategy allotted to them i.e., With such a strategy, no player, after learning the moves of all his opponents, cannot come up with something better, provided the opponents do not change their strategies

Nash and the Prisoner’s Dilemma P2 cooperates P2 defects P1 cooperates 3, 30, 5 P1 defects 5, 01, 1

Nash and Chicken P2 cooperates P2 defects P1 cooperates 3, 32, 4 P1 defects 4, 21, 1

Game Balance An unbalanced game is ugly and unsatisfying Important to avoid wasted development on features that are never chosen Aesthetic purity Marriage of design and function

Three Types of Game Balance Player/Player  Each player gets no special advantage but their skill Player/Gameplay  Learning curve is match with rewards to keep player playing Gameplay/Gameplay  Features within game must be balanced against each other

Player/Player Balance Half the fun of games like Virtua Fighter is seeing how different fighting styles compete with each other If all the characters have the same moves, the game would be rather dull Can Sarah beat Lion every time? If so, it’s not terribly unbalanced unless a beginner playing Sarah consistently beats an expert playing Lion, and even then, it may not be not critical if there is a large range of characters to choose from

Player/Player Balance Victory should be achieved by skill and good judgment This doesn’t mean there shouldn’t be an element of luck Most strategies involve a gamble Deciding whether a risk is worth taking is part of the fun for many people Random elements should not favor just one player

Symmetry The simplest way to ensure perfect balance is by exact symmetry Not only symmetrical in weapons, maneuvers, hit points etc., but symmetrical in level (i.e. no player starts with a better position) Although a fair solution, it is rarely interesting

Symmetry Symmetric maps would look unrealistic, and is a too obvious solution to equalize the odds Better to have a level which is functionally symmetrical, but not obviously Have players be flanked by different geographical barriers, needing different units to proceed. The tough (but best!) solution is to give each player different choices, but giving them the same chance to succeed

Symmetry If players are able to choose their starting positions, then you don’t want any position on the map to have an overwhelming advantage Most general solution in this scenario is to avoid making the initial setup important (e.g. there’s resources everywhere!)

Symmetry Remember, only games should be fair If you’re making an historical simulation, then balance is a less important issue In a Conquest of Mexico simulation, it would be terribly unbalanced to have one player be the Conquistadors, and the other play the Aztecs

Player/Gameplay Balance “There is not a university in the world that I am aware of where in order to graduate with a Computer Science degree, you need to have written a program that is used by another individual, much less be graded on your ability to do so.” - Bill Buxton, Chief Scientist, Alias|Wavefront

Player/Gameplay Balance Sometimes developers get so enveloped in implementing “nifty ideas” and coding bells and whistles, that they forget that people will actually play it! Think about the player’s relationship with the game

A Bad Example (that will be offensive to some) “By using the plus and minus keys next to each trait on the menu, you can take points away from some traits and add them to others to get the balance you want. If you really don’t like the hand you’ve been dealt, you can click REROLL to get a different set of values for the various traits.”  The Baldur’s Gate Official Strategy Guide

Player/Gameplay Balance Balance challenges along the player’s learning curve RPG’s – don’t just make the monsters tougher as I gain experience – give me more options and abilities! Reward the player Let the machine do the work Make a game you don’t have to play against

Reward the Player Players will make mistakes in the beginning In order to keep encouraging the player to continue playing, give him a reward for learning something new and applying it properly Gameplay payoff and graphics need to be worthwhile Widen the gaming experience! “Now that I can do the flying scissors kick, I see a whole new use for the reverse punch!”

Let the Machine do the Work If the game involves tedious tasks that aren’t fun, don’t make the player do them Often a question of interface Don’t bother the player – make the AI do it Some designers cross the line between gameplay feature and chore Example: RPG’s that come with graph paper for you to map the dungeons

Don’t Play Against the Game Player should succeed with skill and judgment, not because he goofed up so many times that there’s only one possible solution left. Some games are designed around the need to save – BAD BAD BAD!!! A game that requires reloading as a normal part of the player’s progress is fundamentally flawed.

Gameplay/Gameplay Balance We want there to be a variety of interesting choices rather than a single choice that always dominates This isn’t easy to establish because the optimum choices depend on the choices other players make It’s not easy to see how frequently different choices will be worth making, but this must be known in order to balance the game

Intransitive Relationships transitive, adj. - being or relating to a relation with the property that if the relation holds between a first element and a second and between the second element and a third, it holds between the first and third elements intransitive, adj. – not transitive (duh)

Intransitive Game Mechanics Consider a SF II style game with three main attacks. Forward kick, stomp, and leg sweep Leg sweep beats forward kick Forward kick beats stomp Stomp beats leg sweep Sound familiar?

Intransitive Game Mechanics Against an AI that chooses randomly, you can equal its score by continually executing one move (e.g. leg sweeps). You will win, lose, and draw 1/3 of the time A human opponent would recognize this behavior, and adapt by using more stomps, which would force me to use more forward kicks, etc.

The Interaction Matrix Leg SweepForward KickStomp Leg Sweep0+1 Forward Kick0+1 Stomp+10 Shows payoff for playing a maneuver vs. your opponent’s maneuver Game is zero-sum

What if the Costs were Different? Suppose a stomp costs 3 points, a forward kick costs 2 points, and a leg sweep costs 1 point Also suppose that by beating your opponent, you gain 5 points, and you lose 5 points if you are defeated The net payoff matrix now becomes…

Net Payoff Matrix Leg SweepForward KickStomp Leg Sweep0+6-3 Forward Kick-60+6 Stomp+3-60 Thus, if I choose leg sweep and you choose stomp, you spend 3 points and I spend 1 point, meaning the difference is +2 points in my favor. But because stomp beats leg sweep, I lose 5 points, netting me -3

Finding the Ratio of Use We’ll call the net payoff for using each move L, F, and S. We’ll call the respective frequencies l,f, and s. Thus, the net payoff for using the leg sweep is: L = (0 x l) + (6 x f) + (-3 x s) These values are taken from the net payoff matrix

The Equations L = 6f – 3s F = 6s – 6l S = 3l – 6f Since it’s zero-sum: L + F + S = 0 Since we’re looking for the equilibrium: L = F = S = 0

The Equations Solving the equations gives us the ratio: l:f:s = 2:1:2 What this means is that for the game to reach equilibrium, the leg sweep and stomp need to be used 40% of the time, while the forward kick is used 20% of the time This isn’t immediately obvious, hence the need to do the math If one option is expensive, often the other options are most affected

Odd-Number Intransitive Relationships SamuraiShugenjaAshigaruArcherNinja Samurai0+1 Shugenja0+1 Ashigaru 0+1 Archer Ninja+1 0

Or… Samurai Shugenja Ashigaru Archer Ninja

Even-number Intransitive Relationships ArcherWarriorBarbarianSorcerer Archer0+10 Warrior0+1 Barbarian+10 Sorcerer0+10 Some players find this asymmetry appealing, since the player doesn’t merely have to learn a cyclical pattern of win-lose relationships

Or… Archers Sorcerers Barbarian Warrior =

The Three Magic Rules of Balance Player/Player – A player should never be put in an unwinnable situation through no fault of their own Player/Gameplay – The game should be as fun to learn as it is to play, and it should be more fun the more you master it Gameplay/Gameplay – All options must be worth using sometimes, and the net cost of using each option must be proportional with the payoff you get for using it