喆学（194）《优化付费订阅的免费多人游戏》设定与初步分析

摘要：This post will introduce the setting and preliminary analysis of the journal article "Optimizing Paid Subscriptions for Free Multi

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“喆学（194）：精读期刊论文

《优化付费订阅的免费多人游戏》

设定与初步分析”

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"Zhexue (194): Intensive reading of journal articles

“Optimizing free-to-play multiplayer games with premium subscription”

Setting and Preliminary Analysis"

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本期推文将从思维导图、精读内容、知识补充三个方面介绍精读期刊论文《优化付费订阅的免费多人游戏》设定与初步分析。

This post will introduce the setting and preliminary analysis of the journal article "Optimizing Paid Subscriptions for Free Multiplayer Games" from three aspects: mind map, in-depth reading content, and supplementary knowledge.

一、思维导图（Mind Maps）

二、精读内容（Intensive reading content）

这款多人免费游戏市场总规模为1，活跃玩家数n在0到1之间。核心内容免费，另设付费订阅，标价每单位时间p，可获持续更新的高级道具。所有玩家每单位时间获基础效用u，付费者额外得道具效用Ui。游戏把玩家分入若干对局，每局十到百人，彼此可见装备。依据社会比较理论，付费者看到非付费者时每单位时间增效用，非付费者见付费者每单位时间减效用，同阶层无比较。玩家体验还受总人数影响。

This free-to-play multiplayer game has a total market size of 1, with the number of active players n ranging from 0 to 1. Core content is free, with a paid subscription priced at p per unit of time, granting access to continuously updated high-level items. All players gain a base utility u per unit of time, with paying players receiving additional item utility Ui. The game divides players into several matches, each with ten to one hundred players, and players can see each other's equipment. Based on social comparison theory, paying players gain increased utility per unit of time when seeing non-paying players, while non-paying players lose decreased utility per unit of time when seeing paying players; there is no comparison within the same social stratum. Player experience is also affected by the total number of players.

玩家体验随总人数变化，人多排队快，人少等待久，单位时间平均摩擦成本与人总数成反比。每局随机抽人，免费者单位时间效用等于基础值减去因见付费者而生的不快再减去摩擦成本，付费者等于基础值加向下比较带来的满足减去订阅费再减摩擦成本，其中付费者比例在零到一之间，社会比较效应不超过基础效用，道具直接效用简化为零，仅聚焦比较心理。

Player experience varies with the total number of players; more players mean faster queues, fewer players mean longer wait times, and the average friction cost per unit time is inversely proportional to the total number of players. Players are randomly selected each round. For free players, the unit time utility equals the base value minus the unpleasantness caused by seeing paying players, plus the friction cost. For paying players, the unit time utility equals the base value plus the satisfaction derived from downward comparison minus the subscription fee, minus the friction cost. The proportion of paying players is between zero and one. The social comparison effect does not exceed the base utility, and the direct utility of items is simplified to zero, focusing solely on the psychology of comparison.

两类玩家的效用差决定付费者比例，而比例又反过来影响效用，三者须同时满足均衡条件。现实中玩家信息有限，未必总能最优选择，故只关注能经受小幅扰动并自行恢复的稳健均衡，类似演化博弈中的稳定概念。

The difference in utility between the two types of players determines the proportion of paying players, and this proportion, in turn, affects utility. All three must simultaneously satisfy the equilibrium condition. In reality, players have limited information and may not always make the optimal choice. Therefore, we only focus on robust equilibria that can withstand small perturbations and recover on their own, similar to the concept of stability in evolutionary game theory.

稳定均衡分三种：全免费时无人愿付费、全付费时无人愿免费，或内部比例下两类效用恰好相等且轻微扰动后能复原。若存在多个稳定点，玩家倾向选整体效用最高的那一个。研究按向下比较效应大于或小于向上比较效应两种情形分别求解均衡比例，并指出二者差异显著，不同群体或游戏可据此匹配策略。

There are three types of stable equilibria: no one is willing to pay when everything is free; no one is willing to give things away when everything is paid; or the two types of utility are exactly equal under the internal proportion and recover after a slight perturbation. If multiple stable points exist, players tend to choose the one with the highest overall utility. This study solves for the equilibrium proportions separately for two scenarios: the downward comparison effect is greater than or less than the upward comparison effect, and points out that the differences between the two are significant. Different groups or games can use this to match strategies.

当向下比较效应强于向上比较效应时，若订阅价很低，全部玩家会转为付费；价格很高则全部免费；价格居中，出现内部均衡，付费比例随价格上升而下降。若向上比较效应更强，则只有全付费或全免费两种边界均衡，无中间状态。内部均衡之所以要求向下比较效应更大，是因为只有付费者对比例变化更敏感，系统才能通过他们向免费迁移来抵消扰动，从而维持稳定。

When the downward comparison effect is stronger than the upward comparison effect, if the subscription price is very low, all players will switch to paying; if the price is very high, everything will be free; if the price is in the middle, an internal equilibrium emerges, and the proportion of paying players decreases as the price increases. If the upward comparison effect is stronger, there are only two boundary equilibria: all paying or all free, with no intermediate state. The internal equilibrium requires a stronger downward comparison effect because only when paying players are more sensitive to changes in the proportion can the system offset disturbances through their migration to free, thus maintaining stability.

三、知识补充（Knowledge supplement）

个体将自己与在某方面表现更好或地位更高的人进行比较，由此产生的心理与行为反应称为向上比较效应。可能的结果：

The upward comparison effect refers to the psychological and behavioral responses an individual experiences when comparing themselves to those who perform better or have higher social status in a certain area. Possible consequences include:

1.激励效应（积极作用）：

1. Motivational effect (positive):

个体可能把他人的成功视为榜样，激发自我提升的动机。例如，学生看到成绩优异的同学后努力学习，以期赶上对方。

Individuals may see the success of others as a role model, motivating them to improve themselves. For example, a student might study harder after seeing a classmate with excellent grades, hoping to catch up.

2.挫败效应（消极作用）：

2. Frustration effect (negative):

若差距过大或个体自我效能感较低，则可能产生自卑、焦虑、嫉妒等情绪，降低自我评价。在网络社交中，人们看到他人展示的“理想生活”时，常产生向上比较，可能激励自我提升，也可能造成“幸福感下降”或“社交媒体焦虑”。

If the gap is too large or the individual's self-efficacy is low, it may lead to feelings of inferiority, anxiety, and jealousy, lowering self-esteem. In online social interactions, when people see others showcasing their "ideal lives," they often engage in upward comparison, which may motivate self-improvement but could also lead to decreased happiness or "social media anxiety."

个体将自己与在某方面表现更差或地位更低的人进行比较，由此产生的心理与行为反应称为向下比较效应。可能的结果：

The downward comparison effect refers to the psychological and behavioral responses an individual experiences when comparing themselves to those who perform worse or have lower social status in a certain area. Possible consequences include:

1.自我提升效应（积极作用）：

1. Self-enhancement effect (positive):

通过看到他人处境更糟而感到安慰，从而提升自我评价与情绪状态。例如，病人看到比自己病情更严重的人，会觉得“我还好”，情绪得到缓解。

Feeling comforted by seeing others in worse situations can improve self-esteem and emotional state. For example, a patient seeing someone with a more severe condition might feel "I'm okay," experiencing emotional relief.

2.自满或防御效应（潜在消极作用）：

2. Complacency or Defensive Effects (Potential Negative Impacts):

若过度依赖向下比较，可能导致自满、不思进取或对问题的否认。

Over-reliance on downward comparison can lead to complacency, lack of motivation, or denial of problems.

向上比较与向下比较并非对立的心理过程，而是个体在社会比较中灵活运用的两种互补性调节机制。人们通常会根据自身的心理状态与情境需求，选择不同的比较方向。当个体渴望提升能力、改善绩效或实现自我超越时，往往倾向于进行向上比较，通过参照更优秀的他人获取榜样激励与学习动力，从而促进个人成长与目标导向行为。而当个体处于压力、挫折或自尊受威胁的状态下，则更可能进行向下比较，通过与境况较差者的对照获得情绪安慰与心理防御，维持积极的自我评价。因此，向上比较与向下比较共同构成了个体在社会情境中调节自我认知与情绪反应的重要机制，二者的动态平衡有助于个体在激励与防御之间实现心理适应。

Upward and downward comparisons are not opposing psychological processes, but rather two complementary regulatory mechanisms that individuals flexibly utilize in social comparison. People typically choose different comparison directions based on their psychological state and situational needs. When individuals aspire to improve their abilities, enhance performance, or achieve self-transcendence, they tend to engage in upward comparisons, seeking inspiration and motivation from role models of more successful individuals, thereby promoting personal growth and goal-oriented behavior. Conversely, when individuals are under stress, experiencing setbacks, or having their self-esteem threatened, they are more likely to engage in downward comparisons, seeking emotional comfort and psychological defense by comparing themselves to those in less favorable circumstances, thus maintaining a positive self-evaluation. Therefore, upward and downward comparisons together constitute an important mechanism for individuals to regulate self-cognition and emotional responses in social situations, and their dynamic balance helps individuals achieve psychological adaptation between motivation and defense.

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翻译：谷歌翻译

参考资料：谷歌、Chat GPT

参考文献：Yunke Mai, Bin Hu. Optimizing free-to-play multiplayer games with premium subscription [J]. Management Science ,2023, 69(6): 3437-3456.