Powerball lottery - Rules and patterns of play

On January 13 2016, the Powerball lottery made the news by reaching a jackpot of $1.5B, the largest in US history. Lotteries are interesting because they are a gambling game used by governments to raise funds without raising taxes. Let's have a closer look at the rules, odds, and profits made by lotteries, using Powerball as an example since it's one of the lotteries with highest sales.

Basics

The official simplified rules stipulate that five white balls are drawn among 69, and one red ball among 26. The jackpot is won by matching all five white balls in any order and the red Powerball. The jackpot starts at $40M, and increases every drawing by at least $10M until won. Combined with the minuscule odds of being won, this mechanic makes it possible for the jackpot to become huge. The match-5 prize, worth $1 million, is won by matching five white balls in any order. Consolation prizes, worth $4 to $50k, are won by matching at least three white balls or at least the red Powerball. Tickets cost $2. Players can buy as many tickets as they want.

Drawings happen twice a week, on Wednesdays and Saturdays. The boxplot below plots the median number of tickets sold per drawing. Data consists of drawings with jackpots below $200M, from April 13 2013 (right after California joined the Powerball group) to January 20 2016. Drawings sell an average of 14.0 million tickets on Wednesdays, versus 15.4 millions on Saturdays (p=.0001).

Jackpots predict sales

Ticket sales are very predictable from the jackpot amount. This is shown in the graph below. For jackpots under $300M, the number of tickets sold increases quadratically with the jackpot. Above $300M, we do not have enough data to build a model, but the trend seems logarithmic, probably because ticket sales grow much faster than the jackpot.

Optional features

Power Play is an optional feature that launched in 2001. As of 2015, it multiplies the value of non-jackpot winnings by 2 to 10 by paying an extra $1 per ticket. A wheel determines the multiplier for each drawing (24/43 chance for 2x down to 1/43 for 10x). Interestingly, Power Play used to have a 1x multiplier, but it was removed within a year. Why have a bonus that is sometimes not a bonus?

Random picks: Players can choose their 6 numbers themselves, or have the computer pick them randomly. About 70% to 80% of purchases are computer picks.

Annuity vs cash

Jackpot winners can choose to receive their share of the jackpot immediately through a 30-year annuity, or immediately in cash. When winners choose the cash, they immediately receive the advertised cash amount, which is usually 60-70% the annuity amount. That is why the lottery organizers advertise the jackpot's annuity, and not its cash.

When winners choose the annuity, they immediately receive 1/30th of the cash amount. The organizers invest the rest into government bonds or securities. Every year for 29 years, they sell 1/29th of the bonds and give the proceeds to the winner, with a 5%-interest per year. Choosing the annuity prevents people from blowing through all their money quickly. Yet only one jackpot winner out of 72 chose the 30-year annuity from 2011 to 2015.

21st Century Game Design - Part I

21st Century Game Design, by Chris Bateman and Richard Boon, 2005.

Part I - Games exist primarily to satisfy the needs of an audience

ch1 - Zen game design

Zen Buddhism can not be learned, it can only be experienced. There is no objective perspective on anything. Hence zen game design's tenets: game design reflects needs + there's no single method to design + there exist methods to game design. These methods are:

• first principles: what you want to do -> game world abstraction -> design -> implementation
• clone and tweak: most common method. existing design -> tweak -> implementation
• meta-rules: goal = provoking debate. meta-rules -> design -> implementation
• expressing technology: in teams without actual game designers. technology -> game implementation
• Frankenstein: art or technical materials -> design -> implementation
• story-driven: narrative -> design -> implementation

Participants in the game project: audience, publisher, producer, programmers, artists, marketing/PR, license holder. Example: saving for causal audience is vital; for hardcore audience, it should not break gameplay; for programmers, it's a technical detail; for producer, it's looking at how other games do it.

ch2 - Designing for the market

The commercial success for a medium clears the way for artistic expression, not the way around

A game design is successful when the target audience is satisfied. This justifies the need for an audience model. Existing models: simple distinction hardcore/casual, distinction by genre (but genres are too vague), EA's model, and ihobo's model.

Simple hardcore/casual distinction

hardcore	casual
plays lots of games	plays few games
game literate	game illiterate
plays for the challenge	plays to relax, kill time, and just for fun
segment can be polarized: many can buy the same title	hard to polarize, diverse and disparate

EA's model:

EA's model take-away: do not ignore hardcores because they are the ones pushing a game to broader segments. Corollary: no TV ads are needed if the game is not made for casuals.

iHobo's model:

Evangelist clusters = gaming press, mainstream press, and the 3 million of hardcores in the world. Target clusters = Testosterone (9M players worldwide), lifestyle (30M), and family (90M) gamers.

Design tools for market penetration (aka demographic game design):

• Looking for good gameplay (ie the game being performance-oriented, with stats, clear goals and victory conditions) vs good toyplay (unorganized). Hardcores are driven by gameplay, but lifestyle and family gamers are driven by both.
• Controls should remain accessible for casuals.
• The minimum play session length is usually expressed in terms of the duration of a level or the time between two save points. For casuals, it should be below 15 minutes, but hardcores do not mind core activities of a game taking at least an hour or two. Ex: a typical DotA match takes 45 to 60 minutes, whereas a (small size) Mine Sweeper can take less than a minute. Nintendo games are also famous for allowing the player to quit at any time and provide core activities of at most a few minutes.
• The average play session length is also lower for casuals: they may complete one level at a time, whereas hardcores can aim at 10 levels per play session.
• Play window: total time spent playing the game. The longer the play window, the longer hardcores will spend evangelizing the game. Therefore, despite most of the players not completing the game, content is crucial! The play window can also be extended by introducing hidden features, higher difficulty levels, variety in characters to play with (to increase replayability), and online PVP (although that only works for Testosterone and hardcore gamers).

Phases of penetration: taking the example of The Sims.

1. Hardcore penetration: the game needs challenge, progress, and depth.
2. Hardcore evangelism: the game needs to appeal to the Lifestyle gamer, easy to reach fun, strong marketing, and a strong license.
3. Casual penetration: the game needs fun, toys, short minimum play session.
4. Casual evangelism: the game needs to get the attention of the mainstream press.

ch3 - Myers-Briggs typology of gamers

Assumption: nature of games people enjoy and frequency of play vary with player personality and reaction to situations. The Myers-Briggs model was developed in the 1940s and indicates how an individual would prefer to react to situations in general. See the Myers-Briggs type frequencies in the US. Four pairs of traits:

Type	Opposite type	Game design
Introversion (50% of pop) think then act, needs private time, 1-to-1 communication and relationships	Extroversion (50% of pop) act then think, likes people, deprived when alone	Most games are played by introverts. Extraverts can take long breaks from the game, so provide a todo list for them when they come back to play, otherwise they'll forget what they had to do in their previous play session. Extraverts like DDR because of its performance aspect.
Sensing (70% of pop) live in the present, apply common sense, based on prior experience, likes clear and concrete info	iNtuition (30% of pop) live in near future, new and imaginative approaches, based on theory, comfortable with fuzzy information, seek for patterns)	Learning and problem solving are frequent gameplay elements in many genres. Learning: in tutorials, S will accept linear series of lessons, but N would rather guess by themselves. Problem solving: S will use trial and error, while N will like to use their lateral thinking skills. Therefore, make lateral thinking puzzles (at most) secondary objectives, or allow the player to progress without having completed all of them. Ex: Super Mario 64 only requires 30 stars to unlock new levels. S want simple and usual mechanics, while N won't mind having to guess the rules and a steep learning curve.
Thinking (30% of women, 60% of men) decide from facts and logic, objective, focus on task, think that conflicts are sometimes unavoidable	Feeling (70% of women, 40% of men) decide from emotion, subjective, focus on consequences to people, wish to avoid conflicts	Clear goals for T. Personal encouragement for F, but T may feel patronized. Solution: useful AND aesthetic/fun items are rewards that will satisfy both T and F. Gathering collectibles give goals to T, but should not be a grind. F are motivated and rewarded when they see their actions have impact on the world or other characters. T enjoy receiving critical feedback (a game over with tips), but F will take it personally. Ex: Zelda gives clear goals (good for T), falling or getting hit results in losing half a heart (and not instant death) and Link has an impact on the game world (good for F).
Judging (55% of pop) plan then move, single task at a time, ahead of deadlines, targets and routines to manage life	Perceiving (45% of pop) plan as you go, multitask, work better before deadline, avoid routine and commitment	J want to beat the game (get all the secret bonuses) and complete objectives. P want to improve their abilities, and enjoy the process. For P, goals completed = feedback that they're on track. Non-linear structure is good for P because if they don't like a level, they can try another and keep progressing. J needs to know what to do to progress. Ex: in Tony Hawk or GTA, players need to collect points (good for J) but they can collect them the way they want (various kinds of skate figures or driving/killing missions or sandbox play, good for P).

TJ vs FP: TJ want challenges to overcome (what most current games provide), FP want easy fun (cf Sims or casual games).

Study hypothesis: hardcore player is a 14-28 year old tech savvy male who plays up to 8 games per month. Supposedly, he plays on his own (hence I), is methodological, goal-oriented enjoys conflicts (T), plays games until completion and looks for perfect score/overachiever (J). Previous quantitative work from the Bartle test by Andreasen showed the average hardcore MMO player is IST. Therefore, let's suppose hardcores are IT. Overall, 15% of women and 35% of men are of type IT.

ch4 - DGD1

DGD1 is intended as a tool to aid in market-oriented game design.

Methods: between 2002 and 2004, ask 408 participants (incl 122 women) to answer a 32-question Myers-Briggs personality test, as well as questions on purchasing and playing habits, and do you consider yourself hardcore, casual, or no idea?. Only look at people who play at least one game per year. Survey advertised on hardcore and casual websites/game portals + university students.

Results: clustering gave a sketchy and incomplete result, and FE and SI dimensions did not help to cluster, but 4 clusters appeared anyway: conqueror (TJ), manager (TP), Wanderer (FP), and participant (FJ). Hypothesis rejected: hardcores are found in E and S (and not only I and T). Still, I and N are higher for hardcores and MMO players than casuals. For each of the four types, twice more respondents reported they were casuals than hardcores.

The DGD1 demographic model

Type	Hardcores Desc	Casuals Desc	Progress	Story	Social
Conqueror	ITJ. Want meaningful challenges, strategies and puzzles, want to complete the game. Want lots of content, try to beat themselves. The game is too easy if they don't die at least a few times. Anger, frustration, boredom, and fiero.	ISTJ. FPS and racing games, they play to compete and win. Rely on genre conventions and do not like deviations from the genre. Fiero (although it's oblivious to them) and schadenfreude in PVP, or in GTA for rampages	Rapid advancement: stats in RPG, better gear in FPS	Focus on plot twists/events, not on characters	Online: vocal hardcores from forums and blogs. They also like to win discussions
Manager	ITP. Strategy and tactics. Winning is less important than mastering the game systems: process-oriented, not goal oriented. Conquerors consider them rivals and targets. Patient. Look for challenging but not impossible. Don't look for hidden features but rather refine their current knowledge. Fiero. Civ series.	ISTP. Want familiar settings and realism. Like construction and management games like SimCity. Hate being stuck even if they suck. Hate interruptions and like smooth difficulty curves.	Steady. Give up if no reliable strategy is found quickly.	Plot, not characters.	None?
Wanderer	INFP. Easy fun and toyplay, not challenges. Variety keeps the fun going. Complete levels in aesthetically pleasing ways. Cf Puzzle Bobble/Bust-a-Move: simple controls, bright colors, and actions with direct and satisfying changes to the environment. See also Mario Party and Super Monkey Ball. Need to be able to give up the current task for another different task. May turn to Conqueror or Manager relatives for help. Emotions: finesse, aesthetics, wonder, awe and mystery, but no fiero.	ENFP. Want to accomplish something in the game world without the need for challenges. Games = way to relax. Feeling of progression or else boredom. Lack of market vectors to reach them [although nowadays there's Facebook]	New toys, colorful and imaginative environments	Emotions. Empathy to characters or investment in world/immersion.	Talk about what they like but avoid arguments
Participant	FJ. Games as social entertainment. Cf DDR, The Sims. Little survey data about this group.		Narrative of group of players	Characters and emotions, but in control of them, not just spectator.	Multiplayer, but must face other players in person, not just online (no MMO)

ch5 - Player abilities

Flow = subjects believe they can complete their activity. Subjects have clear goals and direct and clear feedback. Effortless involvement. Goals should be short-term for participant and conqueror, but long-term for Wanderer and manager because they like to figure out the short-term goals themselves.

Caillois' table of the four categories of play helps understand how flow is related to toyplay. In the table, there really is a continuum between Paidia and Ludus.

The relation between the four play styles of DGD1 and Caillois' categories of games

Conqueror Agon	Manager Agon (Alea tolerated)
Participant Mimicry	Wanderer Mimicry (Alea tolerated)

Caillois' table of the four categories of play

-	Agon (competition)	Alea (chance)	Mimicry (simulation)	Ilinx (vertigo)
Paidia (spontaneous play)	Spontaneous races	Counting out rhymes, coin flipping	Masks and disguisement	Children whirling, swinging
Ludus (structured play)	Sports	Betting, lotteries	Theatre	Skiing, mountain climbing

People with high Myers-Briggs Feeling scores prefer avoiding conflicts, therefore they don't like Agon. They're also more likely to like Mimicry since they focus on people. For example, Wanderers appreciate finesse, which is a component of Mimicry. Ilinx resembles immersion, it appeals to everyone.

Temperament theory gives patterns of behaviors, while Myers-Briggs gives patterns of perception or judgement.

Temperament theory

Temperament	Core needs	Myers-Briggs traits	Skills	% of pop
Rational	Knowledge, competence	NT	Strategic: Think and plan ahead, identify the means to achieve a goal, coordinate actions strategically	10%
Idealist	Unique identity, search for meaning and significance	NF	Diplomatic: Resolve conflicts while recognizing individuality, empathy, find similarities through abstraction	15%
Artisan	Freedom to act and ability to impact	SP	Tactical: Read the current content and manage the situation, work out the next step and take action, improvise to overcome problems	25%
Guardian	Belonging and sense of responsibility/duty	SJ	Logistical: Organizing and meeting needs, optimizing and standardizing, protect and ensure safety	50%

Temperament, Myers-Briggs and DGD1

Type	Myers-Briggs traits	Hardcore temperament trait	Casual temperament trait	Flow provenance	Examples
Conqueror	TJ	strategic	logistical	Capacity to see in advance how to address problems (strategic) and iterate/repeat to improve/optimize the solution (logistical). Willingness to fail and repeat	Production of units in RTS, monsters or bosses with patterns (cf Doom monsters)
Manager	TP	strategic	tactical	Planning ahead (strategic) and reacting to rapidly changing situations (tactical). Hardcores like to get lost in their thoughts, ideally without time limitations. Casuals have flow in the action, and need short-term goals.	RTS have both spontaneous maneuvers and long-term strategies. Civ, Chess or puzzles for hardcores.
Wanderer	FP	diplomatic	tactical	Immersion, explicit short-term goals (tactical). Completion of goals is not a big thing, it happens almost as a side-effect of exploration. Give them time to explore.	Platformers (goal is obvious and challenges relatively easy)
Participant	FJ	diplomatic	logistical	Feeling of belonging, toyplay, optimize relationships (logistical) with other characters or players, immerse themselves in social situation	The Sims, Animal Crossing

Casual audience is best approached with familiar settings and content, and with gameplay that revolves around optimization or thinking on your feet (tactical). Hardcores prefer original games that give them a sense of identity (diplomatic), and problems to solve (strategic), e.g. Final Fantasy focuses on story and strategic battles.

Fundamentals of Game Design, ch 14: Strategy Games

Strategy = planning series of actions against the opponent(s). There's usually a large variety of potential actions. Analysis paralysis happens when players have to wait for the other to be done thinking about his next move. In RTS, decisions have to be taken quickly. The basic mechanics of strategy games involve units and the factories that produce them. There can be diplomacy if the scenario is long enough. The attributes of units can be: (max) health, (max) armor, atk, ammo, aspd, area of effect, range, and others ...

Melee unit's value (eg footman) = max HP * shot power * aspd * accuracy * range * walk speed
Ranged static unit's value (eg tower) = max HP * shot power * aspd * accuracy * range^2/2. Squared because the area defended increases quadratically. Halved because units can generally only cover one side/half, the one side towards the enemy.

Most RTS make little use of line of sight and terrain realism; a unit on top of a hill could see further, a unit behind a hill could be hidden from those behind it, units have atk/def/... bonus when they are in their favorite terrain.

Gold is too important for war for mines to be spread out randomly. Spreading randomly means randomly-chosen players get a random advantage from the start, which is unfair. Mines randomly spread also means that players will hunt for mines instead of fighting strategically against each other. Large mines should be positioned symmetrically if they are decisive. Random mines should be smaller and not be decisive for winning.

Lanchester's laws: 1- In melee combat, army strength is proportional to number of units because each fighter can engage only one enemy at once. 2- In ranged combat, units can concentrate their power on a single enemy. When 1 unit is attacked by 3 archers, its life decreases 3 times faster, and it can only attack one of them, ie 1/3 of the enemies. Therefore, 3 ranged units are 3*3=9 times stronger than a single unit (whether ranged or not).

Imagine the tight and strong negative feedback loop: "wounded units attack weaker". Then players have to play sub-efficiently all the time, and they're going to wage attrition skirmishes, not big epic battles.

Do not block or forbid certain units or skills in a particular level. Instead, make the level so that this unit or skill is very inappropriate. Ex: to prevent the player to use air units, give the opponent lots of anti-air missiles. To prevent the use of iron, set the victory condition to "collect lots of iron".

Upgrades

Upgrades can apply to a single unit (veteran units have +1 dmg), all units of a particular kind (grunts get +2 atk), all units of all kinds (Iron Age = +2 def to everyone), or even the whole mechanics (production speed divided by 2). Upgrades can be unlocked by spending resources, time, achievement, ... The tech tree can be:

• totally open: player spends point in any tree she wants, like in WoW
• constrained: finish one branch to start another
• fixed: once player starts in a tree, only this tree is available for the rest of the game

Production and distribution

Production process should be concrete and visible, while the distribution process should be abstracted. Ex: plant farms for food = concrete food production. No supply lines as long as there are roads = abstract supply distribution. There are different shades and techniques for supply lines: troop caravans between cities in Heroes of Might and Magic IV, trade caravans between cities in Civilization series or Age of Empires, or Armored Personnel Carrier that transports troops and fuel in Advance Wars. Another approach is to have cities within a zone of influence to be supplied: cf the influence zone mechanics of Populous (you can only build in your area of influence), Civilization, and Black and White (you can only exert your god powers within your zone of influence).

AI techniques

• Game tree search, as used in games where there is a finite number of possible moves, such as Chess or Checkers. Problem: the search space increases exponentially, which is a problem for games like Go which have too many possible moves.
• Neural networks: patterns are identified after repeated exposure. Problem: training takes time, the system can hardly be tweaked, and adding new patterns means re-training the whole system.
• FSM: units transition from one state (walking) to another (shooting) because of events (enemy appeared in sight). Problem: the unit can only be in one state at once.
• Hierarchical FSM: FSM of the commander gives orders such as "take this hill" to captains. Captains have an FSM that gives orders to soldiers in the squad, such as "cover fire" or "make diversion". Each soldier has its own FSM as well: "if I see an enemy, run for cover". hFSM can create emergent behavior.

[Literature] Fundamentals of Game Design, ch 9: Gameplay

What makes a game fun

50% from quality assets, UI, and code. 35% tuning and polishing. 10% imaginative level design. 5% innovation. A pinch of luck.
Guidelines

• Gameplay before story and graphics
• get a feature right or leave it out
• player-centric, know your audience
• abstract or automate the boring parts
• be true to your vision, do not add stuff for marketing or in hope of getting more players
• aesthetics matter

Challenges

Gameplay is in the challenges.

Challenge hierarchy: to complete the game, the player has to complete chapters, levels, missions, and atomic challenges. Atomic challenges are what player focuses on at the moment. The game victory condition should be explicit, as well as the atomic challenges. Intermediary challenges can and should be implicit: the fun comes from figuring out how to use the atomic tools to reach the end goal. That's why the player should be rewarded whatever path she took or means she used.

Many simultaneous atomic challenges put stress on the player (e.g. Impossible Super Mario levels). Difficulty = intrinsic skill + stress. Intrinsic skill = level of skill required to overcome a challenge without any time limit. Sudoku is pure skill. Stress = perception of time pressure for a given skill level. Tetris is pure stress. Therefore, to make the game easier, give more time to complete skill-based challenges.

Commonly used atomic challenges:

• Physical coordination (hand-eye): speed, accuracy, timing and rhythm.
• Logic and maths: challenges solved by reasoning power alone. Rubik's cube, puzzles
• Race: time pressure reduces strategic thinking.
• Factual knowledge: Trivial Pursuit, quizzes
• Memory: French Tarot
• Pattern recognition
• Exploration: spatial awareness, locked doors, traps, mazes, teleporters. Should still have challenges, otherwise it's sightseeing.
• Conflict: strategy, tactics, logistics, defend weak units, stealth. Checkers, chess, Heroes of Might and Magic.
• Economic: accumulate resources (RTS), balance a complex economy (SimCity), care for living things (Black and White)
• Lateral thinking: there is no obvious solution.

Actions

Actions are IG events directly and immediately influenced by player's input. Player spends most of her time using actions to overcome atomic challenges. Therefore, particular attention should be given to how actions can overcome basic challenges.

Some actions do not overcome challenges, but provide hooks for unstructured play (honking in a racing game), creation/expression (customizing your avatar), socialization (emotes), and story participation. There are also meta-actions: loading, saving, configurations, exit, ...

Saving

Save files are good for testing and debugging. In general, saving harms immersion and story flow/discovery (player can see different branches by reloading and taking different choices). Always allow the player to save and reload the game. Saving can be implemented through:

• level passwords (levels of Lucky Luke for Gameboy),
• save slots (player can keep copies of his progression and try different things, Doom),
• quick save when pushing a key (Baldur's Gate 2),
• automatic save/checkpoints (mid-level checkpoints in Mario platformer series).

FDG 2011 comments

A few comments from FDG 2011. My points are [inside brackets], the rest comes from their respective authors.

In the last Madden NFL game, people tended to play the same kinds of games and rarely changed to different types/modes.

[The Player Experience workshop had roughly 5/15 papers accepted. 3/5 accepted paper authors were not here. What's the goal of a workshop then?]

Secondary game objectives should support the primary objectives, otherwise they distract, frustrate or make the player lose. Mario side-scrollers coins sometimes make the player go off-path, but the player always has to go from left to right. Player should be told when an objective is secondary.

Minimalist game design (from one of the designers of Osmos): few rules, few mechanics, few choices, deep (combinatorial) complexity (eg NP complete), simple controls, abstract, tightly-coupled systems.

CRPG guidelines: balance should be based on player expertise or avatar skills, not on character level or XP. Avatar background should affect the game. If the avatar is supposed to have lived in the game world for years, then the game should not remind the player of the avatar background, and NPC should not meet the avatar for the first time. [Possible solution: player's avatar could start as a toddler and ages as the player spends more time IG]

Weak ties make social storytelling. [Add to this that being socially active means connecting your weak ties together (you play the go-between). Therefore, in a distant future, there should not be any social storytelling anymore because everybody will be friend (ie strong tie) with everybody. In fact, Dunbar's number says you can't be friend with more than 120 people (the other people are just contacts, not actual friends). Hence, people will keep trying to (re)connect their weak ties together! Cf also Harris and Nardi's paper on collaborative play in WoW]

Mateas: computational media is a research field in which computer science is seen as a tool for expression.

If you want a domain to progress, and the industry does not care about that domain to progress, then you have to go try academia. [Maybe a good approach in many domains, but it does not work in MMOG because building one requires hundreds of man-years]

Frustration arises when player fails to overcome a challenge. Frustration makes players impatient.

[Literature] Game Balance ch7 - Advancement, progression and pacing

My notes from course 7 of the Game Balance class of Summer 2010, by Ian Schreiber.

Progression is ...	in games that are ...
absolute	PvE/single player
relative to other players	PvP/multiplayer

Flow has two problems: different player audiences have different skills, and players learn throughout the game.
Balance = overall game difficulty, does not solve these problems. Balance only matches audience expectation.
Progression/Pacing = keeping the player in the flow zone. As player skill increases, so do challenges. Progression ensures the game ends in the time frame said by the box (1min for arcade, 40h for RPG, etc.). If the game is endless, then progression = end-game rewarding structure(s).

When transitioning from mid- to end- or elder game, the objectives change from progressing to something else. Game designer has to find something for the player to do. Ex: WoW guild raiding or making your house cute in Farmville/Sims. Problems: some players may like the progression game but not the elder game. The power gathered during progression game should be available and enjoyable during elder game.
Tips:

• As playtesters test the game, they become experts => the game gets tuned harder => make the game easier at the end, and/or keep some playtesters for the end.
• Let players adjust the difficulty themselves (more challenging but also more rewarding levels or adjusting the difficulty level at any time). In PVP, difficulty adjustment should be voluntary (handicap, resources at the beginning, ...).

PVE

Perceived difficulty = (game power challenge + game skill challenge) - (player power + player skill), with:

• Game power challenge = stats (doubling opponents HP makes the game harder)
• Game skill challenge = new enemies or better AI, direct challenge to the player's skill (you need to play better) and not a player's power (you need more hit points to win). A game designer can control power-related, but not skill-related components of difficulty.

Large luck component or shallow mechanics: a short increase in player skill as the player masters what little they can at the beginning. Then skill plateau (player is as good as she can ever be). A minute to learn, a minute to master. This is the design of educational games (where skill is not the priority).

Giving practice zones where new weapons or powers are acquired makes players learn/increase their skill faster. Skill gating = progressively harder challenges, guarantee that if players complete a challenge, then they are ready for the next. Skill gating != practice zones.

Psychology: “reward schedule” or “risk/reward cycle”: you don’t just want the players to progress, you want them to feel like they are being rewarded for playing well. Reward not too rarely and not too often. Many small rewards are more efficient than a single big reward. Regular rewards = bad. Reward for something players were looking for (otherwise the game seems too easy) and not for a random event (eg "inflict exactly 123 dmg"). 3 kinds of rewards related to progression: increasing player power, level transitions, and story progression.

Increasing player power

If the most fun toy in your game is only discovered 2/3rds of the way through, that’s a lot of time the player doesn’t get to have fun. How do you actually keep the player engaged when you've given away all the cool toys early in the game? One way is if your mechanics have a lot of depth, you can just present unique combinations of things to the player to keep them challenged and engaged. Warning: this is really hard to do in practice. You can also use other rewards more liberally after you shut off the new toys: more story, more stat increases, more frequent boss fights or level transitions. Also, toy upgrades.
Better shorten the game than have it too long and boring.

Level transitions

Each level takes a little bit longer than the last: fast progression at start engages player into the game, later levels can be longer because player wants to know the end of the plot.

Story progression

Story really IS a reward. There should be a match between story complication/climax curve and the difficulty curve. Ex: tutorial = exposition scene, miniboss = rising action, final boss = final climax. Final boss should not be as demanding on player skill as kill 10 rats.

Pattern: do not reveal the story only during level transition; instead, revealing additional background story immediately after a fight (even an easy one) makes players feel like they earned it. (But do not do that all the time otherwise it becomes predictble!)

PvP

Acquiring more power than opponents = primary reward. PvP has more options to play with than PVE because everything is relative, there's no defined level/stats to reach to be "strong".

negative feedback loops => more power when behind and less power when ahead => best player alternates => depends on opponents, no one is left behind (ex: Mario Kart with dynamic difficulty adjustment).
positive feedback loop => more power brings more power => best stays best => independent of the opponents, game ends faster, bad start is deadly (ex: League of Legends).

	positive sum	negative sum	zero sum
Definition	sum of all player resources increases over time	players lose power over time. Goal = lose power more slowly than opponents.	fixed amount of resources on the table
Example	Catan, Agricola	Chess	Poker
Positive feedback
Negative feedback			Solution: Each player spends time in the lead before one player's final blow ends the game.

When both players have realized who is going to win, the game should end quickly.

[Literature] A theory of Fun for Game Design

Raph Koster. 2004. A Theory of Fun for Game Design. Paraglyph Press.

Quotes taken from Koster's book. I did not try to put them together in a meaningful way, I just copy-pasted those I found most interesting.

ch3: What games are

The only real difference between games and reality is that the stakes are lower with games.
The more formally constructed your game is, the more limited it will be. Long-lasting games integrate variables from outside the magic circle.
Fun from games arises out of mastery.
With games, learning is the drug. Fun is just another word for learning.

ch4: What games teach us

Most of the game designers working professionally today are self-taught.
Games are viewed as frivolity.
Games almost always teach us tools for being the top monkey.
Most games encourage demonizing the opponent. Can we create games that instead offer us greater insight into how the modern world works?
There has not been a topologically different 2d shooter since [the first 2d shooter to have power ups and bosses and scrolling]. Unsurprisingly, the shooter genre has stagnated and lost market share.
Algorithm for innovation: find a new dimension to add to the gameplay

ch5: What games are not

One of the most self-defeating rallying cry in history: "it's just a game"
The part of games that is least understood is the formal abstract system portion of it, the mathematical part of it. Games need to develop this formal aspect of themselves in order to improve.
This is why gamers are dismissive of the ethical implications of games. They do not see "get a blowjob from a hooker then run her over", they see a power up.
Since games are generally about power, control, and those other primitive things, the stories tend to be so as well.
When games and stories are good, you can come back to them repeatedly and keep learning something new.
Getting emotional effects out of games may be the wrong approach. Perhaps a better question is whether stories can be fun in the way games can.
Different kinds of enjoyment:

• fun is the act of mastering a problem mentally
• aesthetic appreciation
• visceral reactions are generally physical and relate to the physical mastery of a problem
• social status maneuvers, intrinsic to our self-image and our standing in a community

Aesthetics is about recognizing patterns, not learning new ones.
Delight does not last. Recognition is not an extended process.
Fun is the feedback the brain gives us when we are absorbing patterns for learning purposes. Real fun comes from challenges that are always at the margin of our ability.
Fun is contextual. The reason why we are engaging in an activity matter a lot.

ch6: Different fun for different folks

Since brains have different strengths and weaknesses, different people will have different ideal games.
People will usually choose to play the games they are good at, that reflect their strength.

ch7: The problem with learning

Players try to find the optimal path to getting to the ultimate goal.
Exploiters are often the most expert players of a game.
Since we dislike tedium, we'll allow unpredictability, but only in the confines of predictable boxes, like games or TV shows. Unpredictability means new patterns to learn, therefore unpredictability is fun.
Those of us who want games to be fun are fighting a losing battle against the human brain because fun is a process and routine is its destination.
If there is not a quantifiable advantage to doing something [reward], the brain will discard it.
Successful games incorporate:

• preparation [choices made before a given challenge], otherwise it's chance. Can the player prepare in different ways and still succeed?
• a sense of space, otherwise it's trivial. Does the environment affect the challenge?
• core mechanics
• challenges [= content] otherwise it's too short. Can the rule set support multiple types of challenges?
• a range of abilities (otherwise it's simplistic). Can the player bring multiple abilities to bear on the challenge? At high levels of difficulty, does the player have to bring multiple abilities?
• abilities should require skill. Not requiring skill from a player should be considered a cardinal sin in game design.

A learning experience require:

• variable feedback (a greater skill should lead to better reward). Are there multiple success states (ie no guaranteed result)?
• high-level players can not get benefits from easy encounters or they will bottom feed.
• failure must have a cost. Player should have the ability to try again (with another preparation round).

ch8: The problem with people

Particular problems and solutions appeal to particular brain types.
Games are not there to fulfill power fantasies.
[The increasing complexity of games within a genre] has led to a priesthood of those who can master the intricacies. Newcomers can not get into them - the barrier for entry is too high.
Designeritis [= being] hypersensitive to patterns in games.
Given the lack of codification and critique of what games are, game designers have instead operated under the more guild-like model of apprenticeship. They do what they have seen work.

ch9: Games in context

The following chart can be applied to any medium.

User goal	Collaborative	Competitive	Solo
Constructive	Community Team game design	Job Commercial game development	Hobby Modding
Experiential	Performance PVE	Sport PVP	Audience Single player games
Deconstructive	Teaching Strategy guide writing	Criticism Hack and cheats	Analysis Writing this book

It's a lot easier to fail to respond to a painting than to fail to respond to a game.
The closer we get to understand the basic building blocks of games (...) the more likely we are to achieve the height of art.

ch10: The ethics of entertainment

For games to really develop as a medium, they need to further develop the ludemes, not just the dressing. By and large, however, the industry has spent its time improving the dressing [...] It's just easy relative to the true challenge.
The best test of a game's fun [is] playing the game with no graphics, no music, no sound, no story, no nothing.
Ethical questions [in games] are aimed at the dressing.
The ludemes themselves can have social values.
As a medium, we have to earn the right to be taken seriously.

ch11: Where games should go

For games to really step up to the plate, they need to provide us with insights into ourselves.
When you feed a player [with a game], right now, we only know "fun" and "boring". Mastery of the medium of games will have to imply authorial intent. The formal systems must be capable of invoking desired learning patterns.

ch12: Taking their rightful place

Games need to develop a critical vocabulary so that understanding of our field can be shared.
Games will never be mature so long as designers create them with complete answers to their own puzzles in mind.

Epilogue

The challenge game designers face is "how do we create games that do not have one right answer?"
[Game designers] are not geeks in the basement rolling funny-shaped dice. Games deserve respect.