The research found is phrased in the converse, that is, does anonymity increase risky behaviour. The research found also equates risky behaviour with anti-social behaviour which may, or may not, apply to the example of making transparent casino wins and losses.
Mathes and Guest (1976) concluded that anonymity, either by disguise or by being one member of a larger group, does contribute to the willingness to engage in anti-social (risky) behaviour. Ellison-Potter and Deffenbacher (2001) made similar findings and concluded that Participants drove more aggressively when they were anonymous.
Chui (2014) differentiated the above research and found that anonymity itself does not cause antisocial behaviour, however, it does appear to facilitate antisocial behaviour for individuals predisposed to it. D’Agata and Kwantes (2020) hypothesized that feelings of disinhibition in the online realm would be strongly related to engaging in risky behaviors. They also concluded that anonymity does not cause antisocial behaviour but facilitated risky behaviour for individuals with certain personality traits.
Your excellent question has relevance for public policy! In particular would the privity of individuals be outweighed by the desire to avoid the negative individual, family and societal outcomes of gambling.
References
Mathes, E.W. and Guest, T.A. (1976) Anonymity and Group Antisocial Behavior, The Journal of Social Psychology, 100:2, 257-262, https://doi.org/10.1080/00224545.1976.9711936
Ellison-Potter, P., Bell, P. and Deffenbacher, J. (2001), The Effects of Trait Driving Anger, Anonymity, and Aggressive Stimuli on Aggressive Driving Behavior. Journal of Applied Social Psychology, 31: 431-443. https://doi.org/10.1111/j.1559-1816.2001.tb00204.x
Chui, R. (2014), A Multi-faceted Approach to Anonymity Online: Examining the Relations between Anonymity and Antisocial Behaviour. Journal of Virtual World Research. 7:2 https://doi.org/10.4101/jvwr.v7i2.7073
D’Agata, M.T., and Kwantes, P.J. (2020) Personality Factors Predicting Disinhibited and Risky Online Behaviors. Journal of Individual Differences 2020 41:4, 199-206 https://doi.org/10.1027/1614-0001/a000321