[Literature] Game Balance ch10 - Economics and Multiplayer Dynamics

My notes from course 10 of the Game Balance class of Summer 2010, by Ian Schreiber.

Economy

In an economic system, players can generate, consume/destroy, or trade resources in a zero-sum fashion (what a player consumes is what another can not consume). Supply and demand curves:

• supply = f(price) is monotonic (increasing): if I sell for $5, then I also can sell for $10
• demand = f(price) is decreasing: if I accept to buy for $10, I also accept to buy for $5

Market price = where the 2 curves intersect. Prices fluctuate as players need or produce items. Prices fluctuate more if there are fewer players. Demand curves affect each other:

• substitutes: SP potion (+ casting Heal spell) is a substitute for HP potion. Increasing supply of X means decreasing demand of Y.
• item sets/collectibles: when together, sword X + shield Y give +10atk bonus. Increasing supply of X means increasing demand of Y.

The price per unit can increase or decrease depending on how many of that unit the player already has. If more units of the same kind brings increasing bonuses, then marginal cost decreases. If each unit costs more and more, the game is more stable and homogeneous.

Demand increases with scarcity. Example in FPS: ammo can be limited (player don't shot all the time, therefore it'd make sense if they died in one shot) or infinite (trigger happy, players should only loose a bit of life when they're hurt). In RTS, limited number of mines to get gold from means shorter games, whereas more and bigger mines means longer games (and more military encounters).

Closed-system = game systems that are self-contained, nothing outside of them can influence what's inside. Somehow, gold farmers open the closed economies of MMOGs, making them harder to design and control. The game should not allow players who have more RL money to have more power, it should just allow more options/variety. Experts using cheap/default CCG decks should beat novices using expensive/rare decks. Yet RL money could be used to speed up progression/avoid grinding.

Inflation happens in MMOs because the game is positive sum (money comes from quests and monsters). New players will never catch up unless negative-sum (sinks) or zero-sum elements are included. Negative-sum/Sinks can be: NPC, repair and death penalties, luxury items, or even tax richest players and give that money to poor players (Robin Hood transforming positive-sum into zero-sum). Zero-sum can be: player-bound/non-tradable items, quests can be repeated but reward only once. Nothing tradable also is a solution.

Trading mechanics

By giving players resources they do not need, they'll have an incentive to trade. Trading mechanics usually serve as a negative-feedback loop, especially within a closed economy. Players are generally more willing to offer favorable trades to those who are behind, while they expect to get a better deal from someone who is ahead (or else they won’t trade at all).

• future agreements: I give X now for Y now and Z later. Players could renege, making them more suspicious when trading with delays.
• scope: powerful resources (eg victory points in Catan) should not be tradable. Tradable resources become more fluid.
• time and phases: players can trade all the time, and trades could take effect at once. Or trading could be limited to certain phases (every 5 turns, or before player starts his building phase) and/or this phase could be timed (eg 5-min timer to bargain).
• evenness: gifts (0 for n) vs even trades (1 for 1) vs uneven trades (1 for n)
• quantification: trading can happen only once per turn/per hour, or as much as player wants. Number of exchanged objects may be bounded as well.
• tax: if trading coalitions are too powerful, put a tax as a cost to trade
• forced: I look at the cards in your hand and pick the best, and give you in exchange my worse card.

Auctions

Auctions = players' willingness to pay for a resource. Auctions work best when the actual cost is variable, different between players, and situational. Each time players decide how much the resource is worth to them, they are making an «interesting choice».

• Many types of auctions (increasing, blind, decreasing, ...),
• with different kinds of rewards (winner gets the entire lot, or first pick in the draft, and/or looser gets bonus or penalty),
• different payers (top bidder, top 2 bidders, top and bottom bidders, all players, ...),
• different recipients (bank = deflation, shared fund, or to other players),
• different events when no one bids (resource given to a random player, or more resources are added to the current resource and the auction restart, or resource is just discarded)

Misc. problems

Name	Problem	Solution
Turtling	Everybody shelters and nobody actually plays because attacking seems more costly and inefficient than defending and waiting for opportunities.	Give incentives to attack (when players wins, she gets more resources next turn), or force players to attack (player has to draw and play one card each turn, and 90% of cards are attack cards).
Kill the leader and sandbagging	Players recognize a clear leader, Players see their best chance to win as eliminating the leader, Players coordinate to attack the leader Players fear to become leader, and play suboptimally	Hidden scores make it impossible to know who is the leader, or make it obvious that eliminating the leader is not the best chance to win, or make the game non-PVP, or make players not able to team-up against another/give advantage when a player defends against many
Kingmaking	One player recognizes they cannot win, Player recognizes that they can give support to any leading player, Losing player chooses one leading player to win	Make players believe they can always win, or make it impossible to know which action will make a king, or make it impossible to choose who to make king, or simply do not allow last players to help leaders
Elimination	Player is killed at the beginning of the game and has to wait for the game to end	Players can only eliminate others if they are strong enough to eliminate all others, or the goal can be to collect points (instead of killing others), or when one player dies, the game stops immediately or within a certain time, and winner is current leader, or make the killed player take control of NPC, or make it interesting to look at other players playing (cf Mafia or Werewolf), or let killed players have goals as well (Cosmic Encounter or BSG)

Final note: balance is not always a must-have. Some games are ostensibly unfair but fun nevertheless. In single-player games, progression matters more than fairness. The unfairness of some one-against-many multiplayer games sometimes makes them fun.

[Literature] Retention in WoW

Thomas Debeauvais et al. 2011. If you build it, they might stay: Retention systems in World of Warcraft. In FDG2011.

I looked at what keeps people playing WoW and which mechanisms retain most effectively which kind of players. Here are my picks from the paper I wrote and sent to FDG2011.

Around 2800 WoW players from Europe, North-America and Asia completed an online questionnaire. Player commitment (and therefore retention) was measured by three metrics: weekly play time, ratio of respondents who have ever stopped playing, and number of years spent playing WoW. All the results mentioned in this article are significant with a p-value below 0.01.

• 96% of respondents have been playing WoW for more than a year, 70% for more than three years.
• 23% of respondents have stopped playing for more than 6 months and have never canceled their subscription - they keep paying even if they do not play!
• On average, people play 23h/week. Asians play more than Westerners. No noticeable difference between men and women.
• Achievement and social actions are motivations that increase the weekly play time. Immersion does not influence the weekly play time.

• Asians are more immersion-oriented than Westerners.
• A higher guild rank (officer or GM > basic member > non-guilded) increases retention.
• Women play with people from real-life more than men.
• People who play with their partner play less than single players, but more than players not playing with their partner. They also stop playing less often.
• 13% of players have found a real-life boyfriend/girlfriend/spouse in WoW.
• There are more players over 45 than players who play more than 40 hours per week (another sample may contain a different ratio, though)

[Literature] Game Balance ch9 - Intransitive mechanics

My notes from course 9 of the Game Balance class of Summer 2010, by Ian Schreiber.

Reminder: “intransitive” is just a geeky way of saying “games like Rock-Paper-Scissors” – that is, games where there is no single dominant strategy, because everything can be beaten by something else. Ex:

• fighting games: blocks > throws > normal > blocks
• RTS: fliers > infantry > archers > fliers
• Advance Wars, and several FPS: Tank > Recon > Anti-tank weapon/infantry > Tank
• visible long-range atk unit > visible med range atk unit with radar > invisible short-range atk unit > visible long-range atk unit without radar
• Magic cards: (X/Y = X atk and Y HP). 3/2 > 1/3 > 2/1 with first strike > 3/2.

Intransitive mechanics are more interesting than transitive ones, esp in PvP games. Players can change strategies in mid-game, or bluff. Balancing is achieved by putting a probability ratio on how often each element of the chain will be used: Rock: 50%, Paper: 20%, Scissors: 30% (= 5:2:3).

Solving RPS

1. Normal-form representation: The payoff matrix consists of: my choices in the rows, the opponent's choices in the columns, and my payoffs in the cells. r, p, and s are the probabilities that my opponent throws rock, paper, and scissors. R, P, and S are the expected payoffs for each of my choice when I choose them during the game.
2. The matrix leads to 3 equations of 3 variables (R = s-p, etc ...)
3. If I know what my opponent plays on average, I know r, p, and s. Therefore I can get my payoffs: R, P and S.
4. The best strategy to use is the one with highest payoff.

Theorems from game theory

• Symmetric games have the same optimal strategy for all players. (Opponent's proba of choosing Rock is same as ours)
• All actions worth taking at all have equal payoffs. (R=P=S)
• In symmetric zero-sum games (eg RPS), all actions worth taking at all have a payoff of zero.

Games are solved considering each player plays optimally (unlike the previous section where I knew r, p, and s). As a side note, a symmetric game is represented using an anti-symmetric matrix. Also, note that the opponent's probabilities r, p, and s must always satisfy r + p + s = 1.

A choice is dominated if one of the other choices is always better. In that case, optimally, the dominated choice will never be picked. Hence, the dominated choice can be removed from the list of choices.

Solving "Rock wins count double"

1. Wins by R count as double. The game is still symmetric: payoffs are the same for the 2 players, the matrix remains anti-symmetric.
2. Obtain 3 equations from the payoff matrix. Each of those equations equals to 0 (cf 3rd theorem above).
3. Solve
4. Conclude: P will be played 50% of the time if wins by R count double.

RPS with costs and partial wins

Solving K(night) - A(rcher) - F(lier), the three being units of a RTS game with costs and partially beating each other. For example, a Knight costing 50g will kill an Archer but lose 20% of its life in the process. What differs is only the payoffs matrix. Solving the equations is the same. Result: k:a:f = 14:10:13.

-	k	a	f
K	50-50 = 0	(-50*0.2)+75 = +65	-50
A	-75+(0.2*50) = -65	75-75 = 0	(-75*0.4)+100 = +70
F	+50	-100+(75*0.4) = -70	100-100 = 0

Asymmetric payoffs

1. If player A wins double with R, but not player B, then there are 2 matrices to look at (one per player).
2. The game is not symmetric anymore, therefore R, P, and S are not 0, and they are different for each player (strictly positive for the player with a bonus, strictly negative for players with handicap). Solve A's matrix: r:p:s = 3:5:4 (interestingly, R has highest payoff but is played least often).
3. Solve B's matrix: r:p:s = 4:5:3.
4. With each of the solution, it's possible to get X and Y.

3-player RPS

In the 3-player case, let's assume I score two points when I beat both of the 2 other players (I play S, they play P and P), and I score one point when I am tie with one player and beat the other (I play R, they play R and S). In that normal case, the intuition is 1:1:1 because it's not different from the 2-player RPS. Hence let's say that wins with R count double (but not the losses). Hence, R vs SS gives +4 (instead of +2), and S vs RS gives -2 (instead of -1). The solution is: r:p:s = 3:4:3.