## Posts Tagged ‘**Arrow’s Impossibility Theorem**’

## Review of Power-Up by Matthew Lane

**Title:** Power-Up

**Author:** Matthew Lane

**Publisher:** Princeton University Press

**Copyright:** 2017

**ISBN13:** 978-0-691-16151-8

**Length:** 264

**Price:** $29.95

**Rating:** 94%

*I purchased a copy of this book for personal use.*

I enjoy creative takes on technical subjects that reveal the mechanics behind familar objects. Video games provide hours of entertainment and challenge. Beyond the need for attractive graphics and effective user interfaces, each game designer must decide how to award points, measure the effect of player choices within the game, and provide a balanced environment that maintains game play without sacrificing challenge. In *Power-Up: Unlocking the Hidden Mathematics in Video Games*, Matthew Lane describes how math enters into video game design. His book is an enjoyable read that taught me a lot about the math behind game design.

## From Physics to Friendship

It would be difficult to find an example of a video game that doesn’t use math in some way. Some games allow exploration without awarding points, for example, but the player must still move around the game world to discover what’s next and every new discovery is an implied “score”. As Lane notes, math provides the foundation for almost every game out there. In *Power-Up*, he divides his coverage into nine chapters:

- Game physics
- Repetition in quiz games
- Voting
- Keeping score
- Chase games
- Complexity
- Friendship
- Chaotic systems
- Value of games

The first eight chapters center on a specific math topic, such as the use of equations to model the physics of a game world and the difficulties of assigning points in games such as recent versions of *The Sims* where friendship can matter as much as health and happiness. The final chapter discusses the value of games as a human activity, specifically mentioning games as educational tools and opportunities to gamble, with a mention of early probability calculations designed to divide the pot fairly in an unfinished game.

I have a bit of math background and have studied probability and statistics in some depth, so I was able to follow almost all of the formulas and related discussion fairly easily. Lane takes care to explain the equations’ inputs and, more importantly, meanings so the calculations’ roles within games can be understood without too much trouble. I’ve seen Arrow’s Impossibility Theorem, which proves that it’s impossible to design a voting system that can’t be manipulated through strategic voting, discussed in several publications; I believe Lane explains the phenomenon effectively and makes the logic behind the theorem clear.

## Repetition and Scoring

While there’s too much material to discuss each chapter in depth, I did want to offer more details about the discussions of repetition and complexity in *Power-Up*. I played early versions of the quiz game *You Don’t Know Jack!* when I was young and, as Lane indicates, I started seeing repeat questions after a relatively short time. In Chapter 2, the author shows how having a relatively small question bank suffers in the face of frequent play. The radical solution, not repeating any questions until they have all been used, has its own issues. Various strategies for reducing the repeat rate have been tried, but most center on reducing the probability that a previously used question will be selected again.

For example, if you have a die with the numbers one through six and roll a one, you might want to make the probability of rolling a one again 1/12 instead of 1/6. The problem is that 5/6 + 1/12 = 11/12, which is less than one. As Lane points out, the actual probability of rolling a one again should be 1/11. If you add 1/11 + 10/11 (the probability of rolling any other number is 2/11, multiplied by five), you get 11/11 = 1. This calculation is interesting and a bit counterintuitive, which points out the creativity required to create fair games that are also fun to play.

Lane also goes into some detail on keeping score, describing several different systems for distance traveled games, tile matching games such as *2048*, and puzzle games such as *Angry Birds*. The discussion of *Angry Birds* was quite interesting for me because it overlapped with a friend’s personal experience. My friend Bill had one of the top scores in the world on the original *Angry Birds*, but he was frustrated that some of the reported scores above him on the leaderboard were impossible to achieve. Not because the point counts were too high, but because there was literally no way to accumulate a specific total. Lane discusses this phenomenon, where it’s possible to prove that some totals can’t be reached within a game’s scoring system, in some depth. I enjoyed the discussion and plan to share it with my friend.

## Conclusions

In *Power-Up*, Matthew Lane describes many of the ways that math powers video games. Similar books and articles have provided in-depth coverage of a specific subject, such as physics models, but his is the first to go into detail on such a wide variety of subjects in the same book. I love his choice of topics and believe the depth of each chapter strikes an excellent balance between detail and length. Highly recommended.